4827 lines
230 KiB
Plaintext
4827 lines
230 KiB
Plaintext
COLLECTED PAPERS
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ON
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WAVE MECHANICS
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BY
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E. SCHRODINGER
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PROFESSOR OF' THEORETICAL PHYSICS AT THE UNIVERSITY OF BERLIN
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TOGETHER WITH HIS
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FOUR LECTURES
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ON
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WAVE MECHANICS
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CHELSEA PUBLISHING COMPANY NEW YORK, N. Y.
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THIRD (AUGMENTED) EDITION
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FIRST GERMAN EDITION, LEIPZIG, 1927 SECOND GERMAN EDITION, LEIPZIG, 1928 FIRST ENGLISH EDITION, LONDON AND GLASGOW, 1928 SECOND (UNALTERED) ENGLISH EDITION, NEW YORK, 1978 THIRD (AUGMENTED) ENGLISH EDITION, NEW YORK, 1982
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THIS THIRD ENGLISH EDITION INCORPORATES THE FULL TEXT OF A WORK ORIGINALLY PUBLISHED AS A SEPARATE
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WORK: 'FouR LECTURES ON WAVE MECHANICS.' THIS WAS ORIGINALLY PUBLISHED AT GLASGOW IN 1928 AND REPRINTED AT GLASGOW IN 1929. THE PRESENT WORK IS
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PRINTED ON 'LONG-LIFE' ACID-FREE PAPER.
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COPYRIGHT @, 1982 BY CHELSEA PUBLISHING COMPANY LIBRARY OF CONGRESS CATALOG CARD NUMBER 80-70108 INTERNATIONAL STANDARD BOOK NUMBER 0-8284-1302-9
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PRINTED IN THE UNITED STATES OF AMERICA
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Contents
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PAPERS
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PAQJI:.
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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
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1. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART I . . . . . . . . . . . . 1
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2. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART II . . . . . . . . . . . . 13
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3. THE CONTINUOUS TRANSITION FROM MICRO- TO MACRO-MECHANICS
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41
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4. ON THE REL\TION BETWEEN THE QUANTUM MECHANICS OF HEISENBERG, BORN, AND JORDAN, AND THAT OF SCHR0DINGER....... . 45
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5. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART III........... . 62
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6. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART IV
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102
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7. THE COMPTON EFFECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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8. THE ENERGY-MOMENTUM THEOREM FOR MATERIAL WAVES . . . . . . . . . . . . . . 130
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9. THE EXCHANGE OF ENERGY ACCORDING TO WAVE MECHANICS . . . . . . . . . . 137
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LECTURES
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1. DERIVATION OF THE FUNDAMENTAL IDEA OF WAVE MECHANICS
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FROM HAMILTON'S ANALOGY BETWEEN ORDINARY MECHANICS
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AND GEOMETRICAL OPTICS ....................................... _ 155 2. ORDINARY MECHANICS ONLY AN APPROXIMATION, WHICH NO
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LONGER HOLDS FOR VERY SMALL SYSTEMS ......................... _ 160 3. BOHR'S STATIONARY ENERGY-LEVELS DERIVED AS THE FREQUENCIES
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OF PROPER VIBRATIONS OF THE WAVES............................. _ 163
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V
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VI
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CONTENTS
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4. ROUGH DESCRIPTION OF THE WAVE-SYSTEMS IN THE HYDROGEN ATOM. DEGENERACY. PERTURBATION ............................ 168
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5. THE PHYSICAL MEANING OF THE WAVE FUNCTION. EXPLANATION
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OF THE SELECTION RULES AND OF THE RULES FOR THE
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POLARIZATION OF SPECTRAL LINES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6. DERIVATION OF THE WAVE EQUATION (PROPERLY SPEAKING)
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WHICH CONTAINS THE TIME ...................................... 176 7. AN ATOM AS PERTURBED BY AN ALTERNATING ELECTRIC FIELD . . . . . . . . . . 177 8. THEORY OF SECONDARY RADIATION AND DISPERSION. . . . . . . . . . . . . . . . . . . . . . 181 9. THEORY OF RESONANCE RADIATION, AND OF CHANGES OF THE
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STATE OF THE ATOM PRODUCED BY INCIDENT RADIATION WHOSE FREQUENCY COINCIDES, OR NEARLY COINCIDES, WITH A
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NATURAL EMISSION FREQUENCY .................................... 185 10. EXTENSION OF WAVE MECHANICS TO SYSTEMS OTHER THAN A
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SINGLE MASS-POINT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,, . . . . . . . . . . . 189 11. EXAMPLES: THE OSCILLATOR, THE ROTATOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 12. CORRECTION FOR MOTION OF THE NUCLEUS IN THE HYDROGEN
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ATOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 13. PERTURBATION OF AN ARBITRARY SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 14. INTERACTION BETWEEN TWO ARBITRARY SYSTEMS., ...................... 203 15. THE PHYSICAL MEANING OF THE GENERALIZED lj,-FUNCTION . . . . . . . . . . . . . . 205
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Preface to the First (German) Edition
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REFERRING to these six papers (the present reprint of which is solely due to the great demand for separate copies), a young lady friend recently remarked to the author : " When you began this work you had no idea that anything so clever would come out of it, had you 1 " This remark, with which I wholeheartedly agreed (with due qualification of the flattering adjective), may serve to call attention to the fact that the papers now combined in one volume were originally written orie by one at different times. The results of the later sections were largely unknown to the writer of the earlier ones. Consequently, the material has unfortunately not always been set forth in as orderly and systematic a way as might be desired, and further, the papers exhibit a gradual development of ideas which (owing to the nature of the process of reproduction) could not be allowed for by any alteration or elaboration of the earlier sections. The Abstract which is prefixed to the text may help to make up for these deficiencies.
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The fact that the papers have been reprinted without alteration in no way implies that I claim to have succeeded in establishing a theory which, though capable of (and indeed requiring) extension, is firmly based as regards its physical foundations and henceforth admits of no alteration in its fundamental ideas. On the contrary, this comparatively cheap method of issue seemed advisable on account of the impossibility at the present stage of giving a fresh exposition which would be really satisfactory or conclusive.
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E. SCHRODINGER.
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ZURICH, November 1926.
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Vll
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Publishers' Note
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Tms translation has been prepared from the second edition of the author's Abhandlungen zur Wellenmechanik, published by Johann Ambrosius Barth, 1928. These papers include practically all that Professor Schrodinger has written on Wave Mechanics.
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The translation has been made by J. F. Shearer, M.A., B.Sc., of the Department of Natural PhilQsophy in the University of Glasgow, and W. M. Deans, B.A., B.Sc., late of Newnham College, Cambridge.
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The translators have tried to follow the original as closely as the English idiom would permit. The English version has been read by Professor Schrodinger. Throughout the book Eigenfunktion has been translated proper function, and Eigenwert, proper value. The phrase eine stuckweise stetige Funktion has been translated a s¢ionally continuous function. These equivalents were decided upon after consultation with the author and with several English mathematicians of eminence.
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viii
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Abstract
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(The references are to pages.)
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THE Hamiltonian analogy of mechanics to optics {pp. 13-18) is an analogy to ge,ometrical optics, since to the path of the representative point in configuration space there corresponds on the optical side the light ray, which is only rigorously defined in terms of geometrical optics. The undulatory elaboration of the optical picture (pp. 19-30) leads to the surrender of the idea of the path of the system, as soon as the dimensions of the path are not great in comparison with the wave-length (pp. 25-26). Only when they are so does the idea of the path remain, and with it classical mechanics as an approximation {pp. 20-24, 41-44); whereas for " micro-mechanical " motions the fundamental equations of mechanics are just as useless as geometrical optics is for the treatment of diffraction problems. In analogy with the latter case, a wave equation in configuration space must replace the fundamental equations of mechanics. In the first instance, this equation is stated for purely periodic vibrations sinusoidal with respect to time (p. 27 et seq_.) ; it may also be derived from a "Hamiltonian variation principle" (p. 1 et seq., pp. 11-12). It contains a " proper value parameter" E, which corresponds to the mechanical energy in macroscopic problems, and which for a single time-sinusoidal vibration is equal to the frequency multiplied by Planck's quantum of action h. In general the wave or vibration equation possesses no solutions, which together with their derivatives are one-valued, finite, and continuous throughout configuration space, except for certain special values of E, the proper values. These values form the "proper value spectrum" which frequently includes continuous parts (the "band spectrum", not expressly considered in most formulae: for its treatment see p. 112 et seq.) as well as discrete points (the "line spectrum"). The proper values either turn out to be identical with the " energy levels " ( = spectroscopic "term "-value multiplied by h) of the quantum theory as hitherto developed, or differ from them in a manner which is confirmed by experience. (Unperturbed Keplerian motion pp. 1-12; harmonic oscillator, pp. 30-34; rigid rotator, pp. 35-36; non-rigid rotator, pp. 36-40 ; Stark effect, pp. 76-82, 93-96.) Deviations of the kind mentioned are, e.g., the appearance of non-integral quantum numbers
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ix
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X
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WAVE MECHANICS
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(viz. the halves of odd numbers) in the case of the oscillator and rotator, and further, the non-appearance of the "surplus" levels (viz. those with vanishing azimuthal or equatorial quantum number) in the Kepler problem. Even in these matters the agreement with Heisenberg's quantum mechanics is complete: this can be proved in general (see below and pp. 45-61). For the calculation of the proper values and the corresponding solutions of the vibration equation (" proper functions") in more complicated cases, there is developed a theory of perturbations, which enables a more difficult problem to be reduced by quadratures alone to a " neighbouring " but simpler one (pp. 64-76). To "degeneracy" corresponds the appearance of multiple proper values (p. 11, p. 33 et seq.). Especially important physically is the case where, as, e.g., in the Zeeman and Stark effects, a multiple proper value is split up by the addition of perturbing forces (general case, pp. 69-76 ; Stark effect, pp. 93-96).
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Up till now the function if, has merely been defined in a purely formal way as obeying the above-mentioned wave equation, serving as its object, so to speak. It is necessary to ascribe to if, a physical, namely an electromagnetic, meaning, in order to make the fact that a small mechanical system can emit electromagnetic waves of a frequency equal to a term-difference (difference of two proper values divided by h) intelligible at all, and further, in order to obtain a theoretical statement for the intensity and polarisation of these electromagnetic waves. This meaning, for the general case of a system with an arbitrary number of degrees of freedom, is not clearly worked out until the end of the sixth paper (pp. 120-123 ; a preliminary attempt for the one-electron problem, on p. 60 et seq., turned out incomplete). A definite ,/,-distribution in configuration space is interpreted as a continuous distribution of electricity (and of electric current density) in actua~ space. If from this distribution of electricity we calculate the component of the electric moment of the whole system in any direction in the usual way, it appears as the sum of single terms, each of which is associated with a couple of proper vibrations, and vibrates in a purely sinusoidal manner with respect to the time with a frequency equal to_,_ the difference of the allied proper frequencies (p. 60 et seq., where ,f, is to be replaced by {,. This simplifies the calculation without essentially modifying it). If the wave-length of the electromagnetic waves, associated with this difference frequency, is large compared with the dimensions of the region to which the whole distribution of electricity is practically confined, then, according to the rules of ordinary electrodynamics, the amplitude of the partial moment in question (or, more accurately, the' square of this amplitude multiplied by the fourth power of the frequency) is a measure of the intensity of the light radiated with this frequency, and with this direction of polarisation. The electrodynamic hypothesis concerning ,f,, and the related purely classical calculation of the radiation, are verified by experience, in so far as they furnish the customary selection and polarisation rules for the oscillator, rotator and the hydrogen atom
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Atlt::;T.KAt..J'l'
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XI
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(easy to show from the results of p. 30 et seq_., p. 35 et seq., and of pp. 1-12; cf. p. 101). Further, they also furnish satisfactory intensity relations for the fine structure of the Balmer lines in •an electric field (p. 82 to p. 92). If only one proper vibration or only proper vibrations of one proper frequency are excited, then the electrical distribution becomes static, yet stationary currents may possibly be superimposed (magnetic atoms, p. 123). In this manner the stability of the normal state and its lack of radiation are explained.
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The amplitudes of the partial moments are closely connected with those quantities (" matrix elements "), which determine the radiation, according to the formal theory of Heisenberg, Born, and Jordan. There can be demonstrated a far-reaching formal identity of the two theories (pp. 45-61 ), according to which not only do the calculated emission frequencies and selection and polarisation rules agree, but also the above-mentioned successful results of the intensity calculations are to be credited as much to the matrix theory as to the present one.
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Everything up till now has referred in the first instance only to conservative systems, although some parts have reached their final formulation only in the sixth paper in connection with the treatment of non-conservative systems. For the latter, the wave equation used hitherto must be generalised into a true wave equation, which contains the time explicitly, and is valid not merely for vibrations purely sinusoidal with respect to time (with a frequency which appears in the equation as a proper value parameter), but for any arbitrary dependence on the time (pp. 102-104). From the wave equation generalised in this way, the interaction of the system with an incident light wave can be deduced, and hence a rational dispersion formula
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(pp. 104-117) ; in all this the electrodynamic hypothesis about ip is
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retained. The generalisation for an arbitrary disturbance is indicated (p. 117 et seq.). Further, from the generalised wave equatio_!l an interesting conservation theorem for the "weight function"• iflyi can be obtained (p. 121), which demonstrates the complete justification of the electrodynamic hypothesis frequently mentioned above, and which makes possible the deduction of the expressions for the components of the electric current density, in terms of the ¢,-distribution (p. 122 et seq.).
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Even the systems treated in the first five papers cannot be conservative in the literal sense of the word, inasmuch as they radiate energy; this must be accompanied by a change in the system. Thus there still seems to be something lacking in the wave law for the tf,-function,-corresponding to the "reaction of radiation" of the classical electron theory, which may result in a dying away of the higher vibrations in favour of the lower ones (p. 116). This necessary complement is still missing.
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The form of the theory discussed so far corresponds to classical (i.e. non-relativistic) mechanics, and does not take magnetic fields into consideration. Therefore, neither the wave equation nor the
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XU
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WAVE M..1£UHA.NllJI:;
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components of the four-current are invariant for the Lorentz transformation. For the one-electron problem an immediate relativisticmagnetic generalisation is readily suggested (pp. 118-120; the Lorentzinvariant expressions for the components of the four-current are not given in the text, but they can be got 1 from the "equation of continuity ", which is to be formed in a way quite analogous to that in the non-relativistic case; cf. p. 122). Though this generalisation yields formally reasonable expressions for the wave lengths, polarisations, intensities, and selection in the natural fine structure and in the Zeeman pattern of the hydrogen atom, yet the actual diagram turns out quite wrong, for the reason that "half integers" appear as azimuthal quantum numbers in the Sommerfeld fine structure formula (p. 9 and p. 119; here the results only are given; V. Fock carried out the calculations quite independently in Leningrad, before my last paper was sent in, and also succeeded in deriving the relativistic equation from a variation principle. Zeitschrift fur Physik, 38, p. 242, 1926). A correction is therefore necessary; all that can be said about it at present is that it must have the same significance for wave mechanics as the " spinning electron " of Uhlenbeck and Goudsmit has for the older quantum theory dealing with electronic orbits (p. 63) ; with this difference, however, that in the latter, together with the introduction of the "spinning electron", the half-integral form of the azimuthal quantum number must be postulated ad hoc, in order to avoid serious conflict with experiment even in the case of hydrogen; while wave mechanics (and also Heisenberg's quantum mechanics) necessarily yields b.alves of odd integers (German : Halbzahligkeit), and thus gives a hint, from the very beginning, of that further extension, which under the regime of the older theory was only shown to be necessary by more complicated phenomena, such as the Paschen-Back effect in hydrogen, anomalous Zeeman effects, structures of multiplets, the laws of Rontgen doublets and the analogy between them and the alkali doublets.
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Addition in the second (German) edition : the first and second of the three new papers now added, namely," The Compton Effect" and "The Energy-Momentum Theorem for Material Waves", are contributions to the four-dimensional relativistic form of wave mechanics discussed in the above paragraph. In connection with the first of these papers I should like above all to remark that, as Herr Ehrenfest has pointed out to me, the figure (p. 128) is incorrect : the pair of wave trains represented in the right half of the figure should coincide completely with the pair on the left, in respect of wave length and the orientation of their planes as well as in breadth of interference fringes (the broken lines).-The second paper, that, on " The EnergyMomentum Theorem ", throws a strong light on the difficulties which a merely four-dimensional theory of lfi-waves comes up against, despite the formally beautiful possibilities of development which present themselves here.-In the last paper, on "The Exchange of
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1 Cf. also a paper by W. Gordon on the Compton Effect, Ztschr. J. Phys. 40, p. 117, 1926.
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ABSTRACT
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xiii
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Energy according to Wave Mechanics ", the many-dimensional, nonrelativistic form is again used. This paper is a first attempt to find out whether, with reference to Heisenberg's important discovery of the " quantum mechanics resonance phenomenon ", it should not be possible to regard those very phenomena which seem to be decisive evidence for the existence of discrete energy levels, without this hypothesis, merely as resonance phenomena.
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WA VE MECHANICS
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Quantisation as a Problem of Proper Values (Part I)
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(Annalen der Physik (4), vol. 79, 1926)
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§ I. IN this paper I wish to consider, first, the simple case of the hydrogen atom (non-relativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the notion of " whole numbers ", merely as such, is not introduced. Rather when integralness does appear, it arises in the same natural way as it does in the case of the node-numbers of a vibrating string. The new conception is capable of generalisation, and strikes, I believe, very deeply at the true nature of the quantum rules.
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The usual form of the latter is connected with the Hamilton-Jacobi differential equation,
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(I)
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A solution of this equation is sought such as can be represented as the
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sum of functions, each being a function of one only of the independent
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variables q.
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Here we now put for Sa new unknown tp such that it will appear
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as a poduct, of related functions of the single co-ordinates, i.e. we put
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(2)
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S =K logy,.
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The constant K must be introduced from considerations of dimensions ; it has those of action. Hence we get
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(I')
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H (q, Klf,oayq,) =E.
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Now we do not look for a solution of equation (I'), but proceed as follows. If we neglect the relativistic variation of mass, equation (I')
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can always be transformed so as to become a quadratic form (of y, and
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its first derivatives) equated to zero. (For the one-electron problem
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I
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VV AV .l!.i lVl.l!.i\.i.tl.Al~ 1\.iO
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this holds even when mass-variation is not neglected.) We now seek
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a function y,, such that for any arbitrary variation of it the integra]
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of the said quadratic form, taken over the whole co-ordinate space,1
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is stationary, y, being everywhere real, single-valued, finite, and con-
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tinuously differentiable up to the second order. The quantum conditions
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are repl,aced by this variation -problem.
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First, we will take for H the Hamilton function for Keplerian
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motion, and show that ip can be so chosen for all positive, but only for
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a discrete set of negative values of E. That is, the above variation
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problem has a discrete and a continuous spectrum of proper values.
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The discrete spectrum corresponds to the Balmer terms and the
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continuous to the energies of the hyperbolic orbits. For numerical
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agreement K must have the value h/21r.
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The choice of co-ordinates in the formation of the variational equa-
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tions being arbitrary, let us take rectangular Cartesians. Then (l')
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becomes in our case
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(l")
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(
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ao-yx,)2
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+
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(a-oyy,)
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2 +
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(o-oi'z1)2
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-
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-K2m-2 (E
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+e-r2)
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y,2
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=
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0
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· '
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e = charge, m = mass of an· electron, r2 = x2 + y2 + z2•
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Our variation problem then reads
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(:tr (a:J -~~(E (3) 8J =8fffdxdydz[ +(:r +
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+ t)f•] =0,
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the integral being taken over all space. From this we find in the usual way
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J Jff ~":\E (4) ½8J = df8y,:t-- dx.dy dz 8i/J[,;2y, +
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+ ~)y,] = 0.
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Therefore we must have, firstly,
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(5)
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V2YJ + 2Km2 (E +-e;.2) 'P =0,
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and secondly,
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(6)
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df is an element of the infinite closed surface over which the integral is taken.
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(It will turn out later that this last condition requires us to supplement our problem by a postulate as to the behaviour of 8y, at infinity, in order to ensure the existence of the above-mentioned continuous spectrum of proper values. See later.)
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The solution of (5) can be effected,for example, in polar co-ordinates,
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r, B, cf,, if y, be written as the product of three functions, each only of
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r, of B, or of <p. The met.hod is sufficiently well known. The function
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of the angles turns out to be a surface harmonic, and if that of r be called X, we get easily the differential equation,
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1 I am aware this formulation is not entirely unambiguous.
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QUA.NTlSATlON AND PROPER VALUES-I
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3
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(7)
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d2x +~ dx + (2mE + 2me2 _ n(n + 1)) =0
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dr2 r dr K 2 K 2r
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r2 . X •
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n=0,1,2,3 ...
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The limitation of n to integral values is necessary so that the surface harmonic may be single-valued. We require solutions of (7) that will remain finite for all non-negative real values of r. Now 1 equation (7) has two singularities in the complex r-plane, at r = 0 and r = oo , of which the second is an" indefinite point" (essential singularity) of all integrals, but the first on the contrary is not (for any integral). These two singularities form exactly the bounding points of our real interval. In such a case it is known now that the postulation of the
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.finiteness of x at the bounding points is equivalent to a boundary
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condition. The equation has in general no integral which remains finite at both end points ; such an integral exists only for certain special values of the constants in the equation. It is now a question of defining these special values. This is the jumping-off point of the whole investigation. 2
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Let us examine first the singularity at r = 0. The so-calJed indicial equation which defines the behaviour of the integral at this point, is
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(8)
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p(p -1) +2p -n(n + I) =0,
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with roots
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(8')
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The two canonical integrals at this point have therefore the exponents n and - (n + I). Since n is not negative, only the first of these is of use to us. Since it belongs to the greater exponent, it can be represented by an ordinary power series, which begins with rn. (The other integral, which does not interest us, can contain a logarithm, since the difference between the indices is an integer.) The next singularity is at infinity, so the above power series is always convergent and. represents a transcendental integral function. We therefore have established that:
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The required solution is (e:r.,cept for a constant factor) a single-valued d,e.finite transcendental integral function, which at r = 0 belongs to the
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exponent n. We must now investigate the behaviour of this function at infinity
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on the positive real axis. To that end we simplify equation (7) by tbe substitution
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(9)
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X =ra-U,
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where a is so chosen that the term with l/r2 drops out. It is easy to verify that then a must have one of the two values n, -(n + I). Equation (7) then takes the form,
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1 For guidance in the treatment of (7) I owe thanks to Hermann Weyl. 1 For unproved propositions in what follows, see L. Schlesinger's Differential Equations (Collection Schubert, No. 13, Goschen, 1900, especially chapters 3 and 5).
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4
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WAVE MECHANICS
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(7')
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)u d2U 2(a + 1) dU 2m(E '!_2 =0
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dr2 + r dr + K 2 + r
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•
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Its integrals belong at r = 0 to the exponents 0 and - 2a - I. For the a-value, a =n, the.first of these integrals, and for the second a-value, a= - (n + 1), the second of these integrals is an integral function and leads, according to (9), to the desired so]ution, which is single-valued. We therefore lose nothing if we confine ourselves to one of the two a-va]ues. Take, then,
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(IO)
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a =n.
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Our solution U then, at r =0, belongs to the exponent 0. Equation (7') is called Laplace's equation. The general type is
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(7")
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U"+(80 +~)U'+(Eo+;)U=O.
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Here the constants have the values
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2mE
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2me2
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(11)
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80 =0, 61=2(a+l), E0 =K2 , E1=K2 .
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This type of equation is comparatively simple to handle for this reason : The so-ca11ed Laplace's transformation, which in general leads again to an equation of the second order, here gives one of the first. This allows the solutions of (7") to be represented by comp1ex integrals. The result 1 only is given here. The integral
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(12)
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U = lez'(z-c1)a1 - 1(z-c 2)a.1 - 1dz
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is a solution of (7") for a path of integration L, for which
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(13)
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JL fz [e"(z - c1)"•(z - o2)••]dz = 0.
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The constants c1, c2, ai, a2 have the following values. c1 and c2 are the roots of the quadratic equation
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(14)
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z2 + 80z + E0 =0,
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and
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(14')
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a1=E1+81c1, a2=_E1+81c2.
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C1 - C2
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C1 - C2
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In the case of equation (7') these become, using (11) and (10),
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(14")
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~ C = 1
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+ ✓
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K2'
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C 2
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=
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-
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✓f]-2{m2E'·
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v - ai = K
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me2 2mE
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+
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n
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+
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1'
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me2 a2 = - KV - 2mE + n + I.
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+
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+
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The representation by the integral (12) allows us, not only to
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survey the asymptotic behaviour of the totality of solutions when r
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1 Cf. Schlesinger. The theory is due to H. Poincare and J. Horn.
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(JUA.NTll::>ATlUl~ Al~V rn,vr .r..n Y.11..1....1u.J.!lo-.L
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tends to infinity in a definite way, but also to give an account of this
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behaviour for one defim'.te solution, which is always a much more
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difficult task.
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We shall at first exclude the case where a1 and a2 are real integers. When this occurs, it occurs for both quantities simultaneously, and
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when, and only when,
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m,e2
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(15)
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Kv-2mE=a real integer.
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-2mE
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+
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Therefore we assume that (15) is not fulfilled. The behaviour of the totality of solutions when r tends to infinity
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in a definite manner-we think always of r becoming infinite through real positive values-is characterised 1 by the behaviour of the two linearly independent solutions, which we will call U1 and U2, and which are obtained by the following specialisations of the path of integration L. In each case let z come from infinity and return there along the same path, in such a direction that
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(16)
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lim ezr = 0,
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z ➔ ao
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i.e. the real part of zr is to become negative and infinite. In this way
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condition (13) is satisfied. In the one case let z make a circuit once round
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the point c (solution U and in the other, round c (solution U
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),
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).
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1
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1
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2
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2
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Now for very large real positive values of r, these two solutions
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are represented asymptotically (in the sense used by Poincare) by
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(17)
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u1,_,ec1rr-o.1( - l)0. 1(e21Tio., -I)r(a1){C1 -C2)°"2 - 1,
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{ U2,.._,ec2rr-o.2( - I)o.2(e21rio.2 - l)r(a2)(c2 - C1)0.1 -1,
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in which we are content to take the first term of the asymptotic series of integral negative powers of r.
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We have now to distinguish between the two cases. I. E > 0. This guarantees the non-fulfilment of (15), as it makes the left hand a pure imaginary. Further, by (14"), c1 and c2 also become pure imaginaries. The exponential functions in (17), since r is real, are therefore periodic functions which remain finite. The values of a1 and a2 from (14") show that both U1 and U2 tend to zero like r-n- 1• This must therefore be validforour transcendental integralsolution U, whose behaviour we are investigating, however it may be linearly compounded from U1 and U2• Further, (9) and (10) show that the function X, i.e. the transcendental integral solution of the original equation (7), always ternls to zero like 1/r, as it arises from U through multiplication by rn. We can thus state: The Eulerian differential equation .(5) of our variation problem has, for every positive E, solutions, which are everywhere single-valued, finite, and continuous; and which tend to zero with 1/r at infinity, under continual oscillations. The surface condition (6) has yet to be discussed.
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1 If (15) is satisfied, at least one of the two paths of integration described in the text cannot be used, as it yields a vanishing result.
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6
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WAVE MEC.tiA.NlC8
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2. E < 0. In this case the possibility (15) is not eo ipso excluded, yet
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we wiil maintain that exclusion provisionally. Then by (14") and (17),
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for r ➔ oo , U1 grows beyond all limits, but U2 vanishes exponentially. Our integral function U (and the same is true for x) will then remain
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finite if, and only if, U is identical with U2, save perhaps for a numerical factor. This, however, can never be, as is proved thus : If a closed
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circuit round both points c1 and c2 be chosen for the path L, thereby satisfying condition (13) since the circuit is really closed on the Riemann
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surface of the integrand, on account of a 1 + a 2 being an integer, then it is easy to show that the integral (12) represents our integral function
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U. (12) can be developed in a series of positive powers of r, which
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converges, at all events, for r sufficiently small, and since it satisfies
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equation (7'), it must coincide with the series for U. Therefore U is
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represented by (12) if L be a closed circuit round both points c1 and c2. This closed circuit can be so distorted, however, as to make it appear
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additively combined from the two paths, considered above, which
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be]onged to U1 and U2 ; and the factors are non-vanishing, 1 and e2"il'l1• Therefore U cannot coincide with U2, but must contain also U1• Q.E.D.
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Our integral function U, which alone of the solutions of (7') is
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considered for our problem, is therefore not finite for r large, on the
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above hypothesis. Reserving meanwhile the question of completeness,
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i.e. the proving that our treatment allows us to find all the linearly
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independent solutions of the problem, then we may state:
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For negative values of E which do not satisfy condition (15) our
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variation problem has no solution.
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We have now only to investigate that discrete set of negative
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E-values which satisfy condition (15). a 1 and a 2 are then both integers. The first of the integration paths, which previously gave us the funda-
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mental values U1 and U2, must now undoubtedly be modified so as to give a non-vanishing result. For, since a 1 -1 is certainly positive, the
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point c1 is neither a branch point nor a pole of the integrand, but an ordinary zero. The point c2 can also become regular if a 2 -1 is also not negative. In every case, however, two suitable paths are readily found
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and the integration effected completely in terms of known functions,
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so that the behaviour of the solutions can be fully investigated.
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Let
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(15')
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Vme2
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l ; l = 1, 2, 3, 4 . . .
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K -2mE
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Then from (14") we have
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(14"')
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a1 -1 =l+n, a2 -1 = -l+n.
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Two cases have to be distinguished : l 6= n and l > n.
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(a) l ~ n. Then c2 and c1 lose every singular character, but instead become starting-points or end-points of the path of integration, in order
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to fulfil condition (13). A third characteristic point here is at infinity
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(negative and real). Every path between two of these three points
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yields a solution, and of these three solutions there are two linearly in-
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~U.tt...J.., .1.J.IJ.tt..-1.-1.V..1.., .n....1..,~ .L .L1,V.L .L...1..1.1, , ..._.._~'-'...._....,
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.._
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dependent, as is easily confirmed if the integrals are calculated out. In particular, the transcendental integral solution is given by the path from c1 to c2. That this integral remains regular at r = 0 can be seen at once without calculating it. I emphasize this point, as the actual calculation is apt to obscure it. However, the calculation does show that the integral becomes indefinitely great for positive, infinitely great values of r. One of the other two integrals remains finite for r large, but it becomes infinite for r = 0.
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Therefore when l ~ n we get no solution of the problem. (b) l > n. Then from (14'"), c1 is a zero and c2 a pole of the first order at least of the integrand. Two independent integrals are then obtained: one from the path which leads from z = - oo to the zero, intentionally avoiding the pole; and the other from the residue at the pole. The l.alter is the integral function. We will give its calculated value, but multiplied by r1i, so that we obtain, according to (9) and (10), the
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solution x of the original equation (7). (The multiplying constant is
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arbitrary.) We find
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(18)
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- ( v-2mE), x-f r K ,
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f
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(
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x
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)
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-x
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n
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e
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-xl-n-
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k~o
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1(~ -2x)k
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(l+n l-n
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-
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I
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-
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) k.
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It is seen that this is a solution that can be utilised, since it remains finite for ·an real non-negative values of r. In addition, it satisfies the surface condition (6) because of its vanishing exponentially at infinity. Collecting then the results for E negative :
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For E negative, our variation problem has solutions if, and only if, E satisfies condition (15). Only values smaller than l (and there is always at least one such at our disposal) can be given to the integer n, which denotes the order of the surface harmonic appearing in the equation. The part of the solution depending on r is given by (18).
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Taking into account the constants in the surface harmonic (known to be 2n + I in number), it is further found that :
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The discovered solution has exactly 2n + I arbitrary constants for any permissible (n, l) combination ; and therefore for a prescribed value of l has l2 arbitrary constants.
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We have thus confirmed the main points of the statements originally made about the proper-value spectrum of our variation problem, but there are still deficiencies.
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Firstly, we require information as to the completeness of the collected system of proper functions indicated above, but I will not concern myself with that in this paper. From experience of similar cases, it may be supposed that no proper value has escaped us.
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Secondly, it must be remembered that the proper functions, ascertained for E positive, do not solve the variation problem as originally postulated, because they only tend to zero at infinity as 1/r,
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and therefore olji/or only tends to zero on an infinite sphere as 1/r2•
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Hence the surface integral (6) is still of the same order as St/J at infinity. If it is desired therefore to obtain the continuous spectrum, another condition must be added to the problem, viz. that St/J is to vanish at
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u
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VY 11. V ~ lVl.l~.A.iilAl.~ ll.iO
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infinity, or at least~ that it tends to a constant value independent of the direction of proceeding to infinity ; in the latter case the surface harmonics cause the surface integral to vanish.
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§ 2. Condition (15) yields
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me4
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(19)
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-Ez = 2K2z2.
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Therefore the well-known Bohr energy-levels, corresponding to the Balmer terms, are obtained, if to the constant K, introduced into (2) for reasons of dimensions, we give the value
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(20)
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from which comes
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K=~,
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27T
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(19')
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Our l is the principal quantum number. n + I is analogous to the azimuthal quantum number. The splitting up of this number through a closer definition of the surface harmonic can be compared with the resolution of the azimuthal quantum into an " equatorial" and a ''polar" quantum. These numbers here define the system of nodelines on the sphere. Also the " radial quantum number " l - n - I gives exactly the number of the " node-spheres ", for it is easily established that the function f(x) in (18) has exactly l - n -1 positive real roots. The positive E-values correspond to the continuum of the hyperbolic orbits, to which one may ascribe, in a certain sense, the radial quantum number oo. The fact corresponding to this is the proceeding to infinity, under continual oscillations, of the functions in question.
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It is interesting to note that the range, inside which the functions of (18) differ sensibly from zero, and outside which their oscillations die away, is of the general order of magnitude of the major axis of the ellipse in each case. The factor, multiplied by which the radius vector enters as the argument of the constant-free function f, isnaturally-the reciprocal of a length. and this length is
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(21)
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v -K
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K 2l h2l al
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2mE me2 = 41T2me2 = l'
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where az = the semi-axis of the lth elliptic orbit. ( The equations follow
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from (19) plus the known relation Ei= ;~}
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The quantity (21) gives the order of magnitude of the range of the roots when l and n are small ; for then it may be assumed that the roots off(x) are of the order of unity. That is naturally no longer the case if the coefficients of the polynomial are large numbers. At present I will not enter into a more exact evaluation of the roots, though I believe it would confirm the above assertion pretty thoroughly.
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QUANTISATION AND PROP.EH. VALUES-J.
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§ 3. It is, of course, strongly suggested that we should try to connect the function t/J with some vibration process in the atom, which would more nearly approach reality than the electronic orbits, the real existence of which is being very much questioned to-day. I originally intended to found the new quantum conditions in this more intuitive manner, but finally gave them the above neutral mathematical form, because it brings more clearly to light what is really essential. The essential thing seems to me to be, that the postulation of "whole numbers " no longer enters into the quantum rules mysteriously, but that we have traced the matter a step further back, and found the "integralness" to have its origin in the finiteness and single-valuedness of a certain space function.
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I do not wish to discuss further the possible representations of the vibration process, before more complicated cases have been calculated successfully from the new stand-point. It is not decided that the results will merely re-echo those of the usual quantum theory. For example, if the relativistic Kepler problem be worked out, it is found to lead in a remarkable manner to half-integral partial quanta (radial and azimuthal).
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Still, a few remarks on the representation of the vibration may be permitted. Above all, I wish to mention that I was led to these deliberations in the first place by the suggestive papers of M. Louis de Broglie,1 and by reflecting over the space distribution of those "phase waves ", of which he has shown that there is always a whole number, measured along the path, present on each period or quasi-period of the electron. The main difference is that de Broglie thinks of progressive waves, while we are led to stationary proper vibrations if we interpret our formulae as representing vibrations. I have lately shown 2 that the Einstein gas theory can be based on the consideration of such stationary proper vibrations, to which the dispersion law of de Broglie's phase waves has been applied. The above reflections on the atom could have been represented as a generalisation from those on the gas model.
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If we take the ~eparate functions (18), multiplied by a surface harmonic of order n, as the description of proper vibration processes, then the quantity· E must have something to do with the related frequency. Now in vibration problems we are accustomed to the "parameter" (usually called .\) being proportional to the square of the frequency. However, in the first place, such a statement in our case would lead to imaginary frequencies for the negative E-values, and, secondly, instinct leads us to believe that the energy must be proportional to the frequency itself and not to its square.
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The contradiction is explained thus. There has been no natural zero level laid down for the "parameter" E of the variation equation (5), especially as the unknown function if, appears multiplied by a function of r, which can be changed by a constant to meet a corresponding
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1 L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925. (Theses, Paris, 1924.) 1 Phy8ik. Zt,'lchr. 27, p. 95, 1926.
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WAVE MECHANICS
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change in the zero level of E. Consequently, we have to correct our anticipations, in that not E itself-continuing to use the same terminology-but E increased by a certain constant is to be expected to be proportional to the square of the frequency. Let this constant be now very great compared with all the admissible negative E-values (which are already limited by (15)). Then firstly, the frequencies will become real, and secondly, since our E-values correspond to only relativ.ely small frequency differences, they will actually be very approximately proportional to these frequency differences. This, again, is all that our " quantum-instinct " can require, as long as the zero level of energy is not fixed.
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|
The view that the frequency of the vibration process is given by
|
|
(22)
|
|
where O is a constant very great compared with all the E's, has still another very appreciable advantage. It permits an understa.nding of the Bohr frequency condition. According to the latter the emission frequencies are proportional to the E-differences, and therefore from (22) also to the differences of the proper frequencies v of those hypothetical vibration processes. But these proper frequencies are all very great compared with the emission frequencies, and they agree very closely among themselves. The emission frequencies appear therefore as deep "difference tones" of the proper vibrations themselves. It is quite conceivable that on the transition of energy from one to another of the normal vibrations, something_:_J mean the light wavewith a frequency allied to each frequency d~fference, should make its appearance. One only needs to imagine that the light wave is causally related to the beats, which necessarily arise at each point of space during the transition; and that the frequency of the light is defined by the number of times per second the intensity maximum of the beat-process repeats itself.
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|
It may be objected that these conclusions are based on the relation (22), in its approximate form (after expansion of the square root), from which the Bohr frequency condition itself seems to obtain the nature of an approximation. This, however, is merely apparently so, and it is wholly avoided when the 1"elativistw theory is developed and makes a profounder insight possible. The large constant O is naturally very intimately connected with the rest-energy of the electron (mc2). Also the seemingly new and independent introduction of the constant h (already brought in by (20)), into the frequency condition, is cleared up, or rather avoided, by the relativistic theory. But unfortunately the correct establishment of the latter meets right away with certain difficulties, which have been already alluded to.
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|
It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes over from one form of vibration to another, than to think
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\:lU.l"1..J.., .1..LUI1..L.LV..Ll 4'..&...._,....., -L ........ _ ..... _ _ ,..
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. --- -
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of a jumping electron. The changing of the vibration form can take place continuously in space and time, and it can readily last as long as the emission process lasts empirically (experiments on canal rays by W. Wien); nevertheless, if during this transition the atom is placed for a comparatively short time in an electric field which alters the proper frequencies, then the beat frequencies are immediately changed sympathetically, and for just as long as the field operates. It is known that this experimentally established fact has hiiherto presented the greatest difficulties. See the well-known attempt at a solution ·by Bohr, Kramers, and Slater.
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Let us not forget, however, in our gratification over our progress in these matters, that the idea of only one proper vibration being excited whenever the atom does not radiate--if we must hold fast to this idea-is very far removed from the natural picture of a vibrating system. We know that a macroscopic system does not behave like that, but yields in general a pot-pourri of its proper vibrations. But we should not make up our minds too quickly on this point. A pot-pourri of proper vibrations would also be permissible for a single atom, since thereby no beat frequencies could arise other than those which, according to experience, the atom is capable of emitting oe,casionally. The actual sending out of many of these spectral lines simultaneously by the same atom does not contradict experience. It is thus conceivable that only in the normal state (and approximately in certain "meta-stable" states) the atom vibrates with one proper frequency and just for this reason does not radiate, namely, because no beats arise. The stimulation may consist of a simultaneous excitation of one or of several other proper frequencies, whereby beats originate and evoke emission of light.
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|
Under all circumstances, I believe, the proper functions, which belong to the same frequency, are in general all simultaneously stimulated. Multipleness of the proper values corresponds, namely, in the language of the previous theory to degeneration. To the reduction of the quantisation of degenerate systems probably corresponds the arbitrary partition of the energy among the functions belonging to one proper value.
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|
Addition at the proof correction on 28.2.1926.
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|
In the case of conservative systems in classical mechanics, the variation problem can be formulated in a neater way than was previously shown, and without express reference to the Hamilton-Jacobi differential equation. Thus, let T (q, p) be the kinetic energy, expressed as a function of the co-ordinates and momenta, V the potential energy, and dT the volume element of the space, " measured rationally ", i.e. it is not simply the product dq1 dq2 dq3 • • • dqn, but this divided by the square root of the discriminant of the quadratic form T (q, p). (Cf. Gibbs' Statistical Mechanics.) Then let if, be such as to make the " Hamilton integral "
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n 1-1. V ~ ll'l~vlllU'il\..ii:)
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|
(23)
|
|
stationary, while fulfilling the normalising, accessory condition
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|
(24)
|
|
The proper values of this variation problem are then the stationary values of integral (23) and yield, according to our thesis, the quantumlevels of the energy.
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|
It is to be remarked that in the quantity a 2 of (14") we have
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|
essentially the well-known Sommerfeld expression - vB'A + v1f. (Cf.
|
|
Atombau, 4th (German) ed., p. 775.)
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Physical Institute of the University of Zurich. (Received January 27, 1926.)
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Quantisation as a Problem of
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Proper Values (Part II)
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(Annalen der Physik (4), vo1. 79, 1926)
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§ I. The Hamiltonian Analogy between Mechanics and Optics
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BEFORE we go on to consider the problem of proper values for further special systems, let us throw more light on the general correspondence which exists between the Hamilton-Jacobi differential equation of a mechanical problem and the "allied" wave equation, i.e. equation (5) of Part I. in the case of the Kepler problem. So far we have only briefly described this correspondence on its external analytical side by the transformation (2), which is in itself unintelligible, and by the equally incomprehensible transition from the e,q_uating to zero of a certain expression to the postulation that the space integral of the said expression shall be stationary.1
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The inner connection between Hamilton's theory and the process of wave propagation is anything but a new idea. It was not only well known to Hamilton, but it also served him as the starting-point for his theory of mechanics, which grew 2 out of his Optics of Nonhomogeneous Media. Hamilton's variation principle can be shown to correspond to Fermat's Principle for a wave propagation in configuration space (q-space), and the Hamilton-Jacobi equation expresses Huygens' Principle for this wave propagation. Unfortunately this powerful and momentous conception of Hamilton is deprived, in most modern reproductions, of its beautiful raiment as a superfluous accessory, in favour of a more colourless representation of the analytical correspondence.3
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1 This procedure will not be pursued further in the present paper. It was only intended to give a provisional, quick survey of the external connection between the
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wave equation and the Hamilton-Jacobi equation. If is not actually the action
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function of a definite motion in the relation stated in (2) of Part I. On the other hand the connection between the wave equation and the variation problem is of course very real ; the integrand of the stationary integral is the Lagrange function for the wave process.
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1 Cf. e.g. E.T. Whittaker's Anal. Dynamics, chap. xL 3 Felix Klein has since 1891 repeatedly developed the theory of Jacobi from quasioptical considerations in non-Euclidean higher space in his lectures on mechanics. Cf. F. Klein, Jahresber. d. Deutsch. Math. Ver. 1, 1891, and Zeits.J. Math. u. Phys. 46,
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13
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, , ..n. f i:, .i.U..Dvlll\.1'41 ~i:',
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Let us consider the general problem of conservative systems m classical mechanics. The Hamilton-Jacobi equation runs
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(1)
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aa,w: +T(qk, aaqwk I\1+ V(qk)=o.
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W is the action fnnction, i.e. the time integral of the Lagrange function T- V along a path of the system as a function of the end points and the time. qk is a representative position co-ordinate ; T is the kinetic energy as function of the q's and momenta, being a quadratic
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form of the latter, for which, as prescribed, the partial derivatives of W with respect to the q's are written. V is the potential energy. To solve the equation put
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(2) and obtain
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W = - Et + S(q1c),
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(I')
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E is an arbitrary integration constant and signifies, as is known, the
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energy of the system. Contrary to the usual practice, we have let the function W remain itself in (l'), instead of introducing the time-free function of the co-ordinates, S. That is a mere superficial_ity.
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Equation (l') can now be very simply expressed if we make use of
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the method of Heinrich Hertz. It becomes, like all geometrical
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assertions in configuration space (space of the variables qk), especially simple and clear if we introduce into this space a non-Euclidean metric by means of the kinetic energy of the system.
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Let T be the kinetic energy as function of the ve]ocities rjk, not of
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the momenta as above, and let us put for the line element
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(3)
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ds2 = 2T(qk, rjk)dt2•
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The right-hand side now contains dt only externally and represents (since <j!cllt = dqk) a quadratic form of the dqk's.
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After this stipulation, conceptions such as angle between two line elements, perpendicularity, divergence and curl of a vector, gradient of a scalar, Laplacian operation (=div grad) of a scalar, and others, may be used in the same simple way as in three-dimensional Euclidean space, and we may use in our thinking the Euclidean three-dimensional representation with impunity, except that the analytical expressions for these ideas become a very little more complicated, as the line
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vn element (3) must everywhere replace the Euclidean line element. We.
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stipulate, that in what follows, all geometrical statements q-space are
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to be taken in this non-Euclidean sense. One of the most important modifications for the calculation is
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1901 (Ges.-Abh. ii. pp. 601 and 603). In the second note, Klein remarks reproachfully that his discourse at Halle ten years previously, in which he had discussed this correspondence and emphasized the great significance of Hamilton's optical works, had "not obtained the general attention, which he had expected''. For this allusion to F. Klein, I am indebted to a friendly communication from Prof. Sommerfeld. See also Atombau, 4th ed., p. 803.
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"lUAl~ .llD.8..llV.l.~ .ti..l.'UJ .L .J.\.V.1. .J.:J.1.1, , .n..uv..a..:.ou i.a.
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that we must distinguish carefully between covariant and contra• variant components of a vector or tensor. But this complication is not any greater than that which occurs in the case of an oblique set of Cartesian axes.
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The dq/s are the prototype of a contravariant vector. The co-
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efficients of the form 21.', which depend on the q,/s, are therefore of a covariant character and form the covariant fundamental tensor. 2T
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is the contravariant form belonging to 21', because the momenta are
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lmown to form the covariant vector belonging to the speed vector qk,
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the momentum being the velocity vector in covariant form. The left side of (l') is now simply the contravariant fundamental form,
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in which the o-::iW's are brought in as variables. The latter form the
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uqk
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components of the vector,-according to its nature covariant,
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grad W.
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(The expressing of the kinetic energy in terms of momenta instead of speeds has then this significance, that covariant vector components can only be introduced in a contravariant form if something intelligible, i.e. invariant, is to result.)
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Equation (I') is equivalent thus to the simple statement
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(I")
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(grad W)2 =2(E- V),
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or
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(l"')
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jgrad WI= v'2(E - V).
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This requirement is easily analysed. Suppose that a function W, of the form (2), has been found, which satisfies it. Then this function
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can be clearly represented for every definite t, if the family of surfaces
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W =const. be described in q-space and to each member a value of W
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be ascribed. Now, on the one hand, as will be shown immediately, equation
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(l"') gives an exact rule for constructing all the other surfaces of the
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family and obtaining their W-values from any single member, if the latter and its W-value -is known. On the other hand, if the sole necessary data for the construction, viz. one surface and its W-value be given quite arbitrarily, then from the rule, which presents just two
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alternatives, there may be completed one of the functions W fulfilling
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the given requirement. Provisionally, the time is regarded as con-
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stant.-The construction rule therefore exhausts the contents of the
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differential equation; each of its solutions can be obtained from a suitably chosen surface and W-value.
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Let us consider the construction rule. Let the value W0 be given
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in Fig. I to an arbitrary surface. In order to find the surface WO + dW0,
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take either side of the given surface as the positive one, erect the normal at each point of it and cut off (with due regard to the sign of dW0 ) the step
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(4)
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ds=- dWo .
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\i2(E - V)
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.LU
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The locus of the end points of the steps is the surface WO + dW0• Similarly, the family of surfaces may be constructed successively on
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both sides.
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The construction has a double interpretation, as the other side of
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the given surface might
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have been taken as posi-
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tive for the first step.
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This ambiguity does not
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hold for later steps, i.e.
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at any later stage of
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the process we cannot
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change arbitrarily the
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sign of the sides of the
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surface, at which we
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have arrived, as this
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FIG,].
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would involve in general a discontinuity in the
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first differential coefficient of W. Moreover, the two families obtained
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in the two cases are clearly identical ; the W-values merely run in the
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opposite direction.
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Let us consider now the very simple dependence on the time. For
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this, (2) shows that at any later (or earlier) instant t +t', the same group
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of surfaces illustrates the W-distribution, though different W-values
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are associated with the individual members, namely, from each W-value
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ascribed at time t there must be subtracted Et'. The W-values wander,
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as it were, from surface to surface according to a definite, simple law,
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and for positive E in the direction of W increasing. Instead of this,
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however, we may imagine that the surfaces wander in such a way that
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each of them continually takes the place and exact form of the following
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one, and always carries its W-value with it. The rule for this wandering
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is given by the fact that the surface WO at time t + dt must have
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reached that place, which at t was occupied by the surface WO+ Edt.
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This will be attained according to (4), if each point of the surface W0
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is allowed to move in the direction of the positive normal through a
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distance
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Edt
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(5)
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ds V2(E- V).
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That is, the surfaces move with a normal velocity
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(6)
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ds
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E
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u = dt = v2(E - V)'
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which, when the constant Eis given, is a pure function of position.
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Now it is seen that our system of surfaces W = const. can be con-
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ceived as the system of wave surfaces of a progressive but stationary wave motion in q-space, for which the value of the phase velocity at
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every point in the space is given by (6). For the normal construction
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QUANTlSATlU.N A.NlJ .t'.tl.V.t'~.tl. VALU~Q-.1.1
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.I. I
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can clearly be replaced by the construction of elementary Huygens waves (with radius (5)), and then of their envelope. The "index of refraction" is proportional to the reciprocal of (6), and is dependent on the position but not on the direction. The q-space is thus optically non-homogeneous but is isotropic. The elementary waves are " spheres ", though of course-let me repeat it expressly once morein the sense of the line-element (3).
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The function of action W plays the part of the phase of our wave system. The Hamilton-Jacobi equation is the expression of Huygens' principle. If, now, Fermat's principle be formulated thus,
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(7)
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1
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1
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P JP O=o ds=o dsv2(E-V)=oJt12Tdt=~oJt12Tdt,
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u
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E
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E E
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JP 1
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P1
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t,
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t1
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we are led directly to Hamilton's principle in the form given by
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Maupertuis (where the time integral is to be taken with the usual
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grain of salt, i.e. T + V =E =constant, even during the variation).
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The " rays ", i.e. the orthogonal trajectories of the wave surfaces, are
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therefore the paths of the system for the value E of the energy, in
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agreement with the well-known system of eq·uations
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aw
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(8)
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Pk=-,o;;q;k--,
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which states, that a set of system paths can be derived from each special function of action, just like a fluid motion from its velocity potential.1 (The momenta Pk form the covariant velocity vector, which equations (8) assert to be equal to the gradient of the function of action.)
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Although in these deliberations on wave surfaces we speak of velocity of propagation and Huygens' principle, we must regard the analogy as one between mechanics and geometrical optics, and not physical or undulaJ,ory optics. For the idea of " rays ", which is the essential feature in the mechanical analogy, belongs to geometrical optics; it is only clearly defined in the latter. Also Fermat's principle
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can be applied in geometrical optics without going beyond the idea of index of refraction. And the system of W-surfaces, regarded as wave surfaces, stands in a somewhat looser relationship to mechanical motion, inasmuch as the image point of the mechanical system in no wise moves along the ray with the wave velocity u, but, on the
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contrary,
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its
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velocity
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(for
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constant
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E)
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is
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proportional
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to
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!.
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u
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directly from (3) as
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(9)
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v=~: =V2'.f=v2(E- V).
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It is given
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1 See especially A. Einstein, Verh. d. D. Physik. Ges. 19, pp. 77, 82, 1917. The
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framing of the quantum conditions here is the most akin, out of all the older attempts, to the present one. De Broglie has returned to it.
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WA V~ M~CHANICS
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This non-agreement is obvious. Firstly, according to (8), the ~ystem's point velocity is great when grad W is great, i.e. where the W-surfaces are closely crowded together, i.e. where u is small. Secondly, from the definition of W as the time integral of the Lagrange function, W alters during the motion (by (T - V)dt in the time dt), and so the image point cannot remain continuously in contact with the same W-surface.
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And important ideas in wave theory, such as amplitude, wave length, and frequency-or, speaking more generally, the waveform-do not enter into the analogy at all, as there exists no mechanical parallel; even of the wave function itself there is no mention beyond that W has the meaning of the phase of the waves (and this is somewhat hazy owing to the waveform being undefined).
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If we find in the whole parallel merely a satisfactory means of contemplation, then this defect is not disturbing, and we would regard any attempt to supply it as idle trifling, believing the analogy to be precisely with geometrical, or at furthest, with a very primitive form of wave optics, and not with the fully developed undulatory optics. That geometrical optics is only a rough approximation for Light makes no difference. To preserve the analogy on the further development of the optics of q-space on the lines of wave theory, we must take good care not to depart markedly from the limiting case of geometrical optics, i.e. must choose 1 the wave length sufficiently small, i.e. small compared with all the path dimensions. Then the additions do not teach anything new; the picture is only draped with superfluous ornaments.
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So we might think to begin with. But even the first attempt at the development of the analogy to the wave theory leads to such striking results, that a quite different suspicion arises: we know to-day, in fact, that our classioal mechanics fails for very small dimensions of the path and for very great curvatures. Perhaps this failure is in strict analogy with the failure of geometrical optics, i.e. " the optics of infinitely small wave lengths", that becomes evident as soon as the obstacles or apertures are no longer great compared with the real, finite, wave length. Perhaps our classical mechanics is the complete analogy of geometrical optics and as such is wrong and not in agreement with reality; it fails whenever the radii of curvature and dimensions of the path are no longer great compared with a certain wave length, to which, in q-space, a real meaning is attached. Then it becomes a question of searching 2 for an undulatory mechanics, and the most obvious way is the working out of the Hamiltonian analogy on the lines of undulatory optics.
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1 Cf. for the optical case, A. Sommerfeld and Iris Runge, Ann. d. Phys. 35, p. 290, 1911. There (in the working out of an oral remark of P. Debye), it is shown, how the equation of first order and second degree for the phase (" Hamiltonian equation") may be accurately derived from the equation of the second order and first degree for the wave function (" wave equation"), in the limiting case of vanishing wave length.
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z Cf. A. Einstein, Berl. Ber. p. 9 et seq., 1925.
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~Uri..l,.J...J.Ur1..1....1.V.J.., J:.J....&..,LI ..L.&.1,1'\_,I.L..&.....111..&..-
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,,6..._ ........ ....., _ _ ...._.
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.... _
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§ 2. " Geometrical" and " Undulatory" Mechanics
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We will at first assume t,hat it is fair, in extending the analogy, to
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imagine the above-mentioned wave system as consisting of sine waves.
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This is the simplest and most obvious case, yet the arbitrariness, which arises from the fundamental significance of this assumption, must be
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emphasized. The wave function has thus only to contain the time in the form of a factor, sin ( . . . }, where the argument is a Jinear function of W. The coefficient of W must have the dimensions of the reciprocal of action, since W has those of action and the phase of a sine has zero dimensions. We assume that it is quite universal, i.e. that it is not only independent of E, but also of the nature of the
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mechanical system. We may then at once denote it by 121-r. The
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time factor then is
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(10)
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sm. (2~ 1rW + const.) = sm. ( - -21hrE-t + 21rSh(qk) + const. ) .
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Hence the frequency v of the waves is given by
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(11)
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Thus we get ·the frequency of the q-space waves to be proportional
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to the energy of the system, in a manner which is not markedly
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artificial.1 This is only true of course if E is absolute and not, as in
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classical mechanics, indefinite to the extent of an additive constant.
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By (6) and (11) the wave length is independent of this additive constant,
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being
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(12)
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u
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h
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,\ =
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-
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v
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=
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-v-:==2===(==E===-===v==)= '
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and we know the term under the root to be double the kinetic energy. Let us make a preliminary rough comparison of this wave length with the dimensions of the orbit of a hydrogen electron as given by classical mechanics, taking care to notice that a " step " in q-space has not the dimensions of length, but Jength multiplied by the square root of mass, in consequence of (3). ,\ has similar dimensions. We have therefore to divide ,\ by the dimension of the orbit, a cm., say, and by the square root of m, the mass of the electron. The quotient is of the order of magnitude of
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-h ,
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mva
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where v represents for the moment the electron's velocity (cm./sec.). The denominator mva is of the order of the mechanical moment of momentum, and this is at least of the order of 10-27 for Kepler orbits, as can be calculated from the values of electronic charge and mass
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1 In Part I. this appeared merely as an a-pproximate equation, derived from a pure speculation.
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YJ .n.. Y ..L:J .U.1..1.:J\..J.I.J..l"J..J..,_ J.\..Jlv
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independently of all quantum theories. We thus obtain the correct
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order for the limit of the approximate region of validity of classical
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mechanics, if we identify our constant h with Planck's quantum of
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action-and this is only a preliminary attempt.
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If in (6), E is expressed by means of (11) in terms of v, then we
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obtain
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(6')
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hv u = -v---2:==(==h::=v==-==V==)=-·
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The dependence of the wave velocity on the energy thus becomes a particular kind of dependence on the frequency, i.e. it becomes a law of dispersion for the waves. This law is of great interest. We have shown in § I that the wandering wave surfaces are only loosely connected with the motion of the system point, since their velocities are not equal and cannot be equal. According to (9), (11), and (6') the system's velocity v has thus also a concrete significance for the wave. We verify at once that
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(13)
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i.e. the velocity of the system point is that of a group of waves, included within a small range of frequencies (signal-velocity). We find here again a theorem for the "phase waves" of the electron, which M. de Broglie had derived, with essential reference to the relativity theory, in those fine researches,1 to which I owe the inspiration for this work. We see that the theorem in question is of wide generality, and does not arise solely from relativity theory, but is valid for every conservative system of ordinary mechanics.
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We can utilise this fact to institute a much more innate connection between wave propagation and the movement of the representative point than was possible before. We can attempt to build up a wave group which will have relatively small dimensions in every direction. Such a wave group will then presumably obey the same laws of motion as a single image point of the mechanical system. It will then give, so to speak, an equivalent of the image point, so long as we can look on it as being approximately confined to a point, i.e. so long as we can neglect any spreading out in comparison with the dimensions of the path of the system. This will only be the case when the path dimensions, and especially the radius of curvature of the path, are very great compared with the wave length. For, in analogy with ordinary optics, it is obvious from what has been said that not only must the dimensions of the wave group not be reduced below the order of magnitude of the wave length, but, on the contrary, the group must extend in all directions over a large number of wave lengths, if it is to be approximately monochromatic. This, however, must be postulated, since the wave group must move about as a whole with a definite
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1 L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925. (Theses, Paris, 1924.)
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QUANTISATION AND PROPER VALUES-II
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21
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group velocity and correspond to a mechanical system of definite energy (cf. equation 11).
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So far as I see, such groups of waves can be constructed on exactly the same principle as that used by Debye 1 and von Laue 2 to solve the problem in ordinary optics of giving an exact. analytical representation of a cone of rays or of a sheaf of rays. From this there comes a very interesting relation to that part of the Hamilton-Jacobi theory not described in § 1, viz. the well-known derivation of the equations of motion in integrated form, by the differentiation of a complete integral of the Hamilton-Jacobi equation with respect to the constants of integration. As we wi11 see immediately, the system of equations called after Jacobi is equivalent to the statement: the image point of the mechanical system continuously corresponds to that point, where a certain continuum of wave trains coalesces in eq_ual phase.
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In optics, the representation (strictly on the wave theory) of a "sheaf of rays" with a sharply defined finite cross-section, which proceeds to a focus and then diverges again, is thus carried out by Debye. A continuum of plane wave trains, each of which alone would fill the whole space, is superposed. The continuum is produced by letting the wave normal vary throughout the given solid angle. The waves then destroy one another almost completely by interference outside a certain double cone; they represent exactly, on the wave theory, the desired limited sheaf of rays and also the diffraction phenomena, necessarily occasioned by the limitation. We can represent in this manner an infinitesimal cone of rays just as weJl as a finite one, if we allow the wave normal of the group to vary only inside an infinitesimal solid angle. This has been utilised by von Laue in his famous paper on the degrees of freedom of a sheaf of rays. 3 Finally, instead of working with waves, hitherto tacitly accepted as purely monochromatic, we can also allow the frequency to vary within an infinitesimal interval, and by a suitable distribution of the amplitudes and phases can confine the disturbance to a region which is relatively small in the longitudinal direction also. So we succeed in representing analytically a "parcel of energy" of relatively small dimensions, which travels with the speed of light, or when dispersion occurs, with the group velocity. Thereby is given the instantaneous position of the parcel of energy-if the detailed structure is not in question-in a very plausible way as that point of space where all the superposed plane waves meet in exactly agreeing phase.
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We will now apply these considerations to the q-space waves. We select, at a definite time t, a definite point P of q-space, through which the parcel of waves passes in a given direction R, at that time. In addition let the mean frequency v or the mean E-value for the packet be also given. These conditions correspond exactly to postulating that at a given time the mechanical system is starting from a given
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1 P. Debye, Ann. d. Phys. 30, p. 755, 1909.
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2 M. v. Laue, idem 44, p. 1197 (§ 2), 1914.
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3 Loe. cit.
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22
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WAVE MECHANICS
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configuration with given velocity components. (Energy plus direction is equivalent to velocity components.)
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In order to carry over the optical construction, we require firstly one set of wave surfaces with the desired frequency, i.e. one solution of the Hamilton-Jacobi equation (l') for the given E-value. This solution, W, say, is to have the following property: the surface of the set which passes through Pat time t, which we may denote by
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(14)
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W = W0 ,
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must have its norma] at Pin the prescribed direction R. But this is
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still not enough. We must be able to vary to an infinitely small
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extent this set of waves Win an n-fold manner (n =number of degrees
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of freedom), so that the wave normal will sweep out an infinitely small
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i(n -1) dimensiona] space angle at the point P, and so that the frequency wi11 vary in an infinitely small one-dimensional region, whereby
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care is taken that all members of the infinitely small n-dimensional
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continuum of sets of waves meet together at time t in the point P in
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exactly agreeing phase. Then it is a question of finding at any other
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time where that point lies at which this agreement of phases occurs.
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To do this, it will be sufficient if we have at our disposal a solution
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W of the Hami]ton-Jacobi equation, which is dependent not only on
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the constant E, here denoted by a1, but also on (n -1) additional constants a 2, a3 ••• an, in such a way that it cannot be written as a function of less than n combinations of these n constants. For then we can,
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firstly, bestow on a1 the value prescribed for E, and, secondly, define a 2, a3 ••• an, so that the surface of the set passing through the point P has at P the prescribed normal direction. Henceforth we understand
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by ai, a 2 ••• an, these va]ues, and take (14) as the surface of this set, which passes through the point P at time t. Then we consider
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the continuum of sets which belongs to the ak-values of an adjacent
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infinitesimal ak-region. A member of this continuum, i.e. therefore
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a set,, will be given by
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aw aw
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aw
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(15)
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W
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+~d
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ua1
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a1
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+~d
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ua2
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a2
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+
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+ ~-dan = const.
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uan
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for a fixed set of values of da1, da2 • • • dan, and varying constant. That member of this set,, i.e. therefore that single surface, which goes
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through Pat time twill be defined by the following choice of the const.,
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, aw aw Gw) (aw) (15)
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W
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+~d
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ua1
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a1
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+
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...
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+~dan=
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uan
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W0
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+
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-
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a1
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da
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o
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1
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+
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...
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+
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~ dan,
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uan o
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(~!:) where
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, etc., are the constants obtained by substituting in the
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0
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differential coefficients the co-ordinates of the point P and the value t
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~=)· of the time ( which latter really only occurs in
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The surfaces (15') for all possible sets of values of da1, da2 ••• dan,
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QUANTISATION AND PROPER VALUES-II
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23
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form on their part a set. They all go through the point P at time t, their wave normals continuously sweep out a little (n -1) dimensional solid angle and, moreover, their E-parameter also varies within a small region. The set of surfaces (15') is so formed that each of the set.a (15) supplies one representative to (15'), namely, that member which passes through P at time t.
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We will now assume that the phase angies of the wave functions which belong to the sets (15) happen to agree precisely for those representatives which enter the set (15'). They agree therefore at time tat the point P.
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We now ask: Is there, at any arbitrary time, a point where all surfaces of the set (15') cut one another, and in which, therefore, all the wave functions which belong to the sets (15) agree in phase? The answer is : There exists a point of agreeing phase, but it is not the common intersection of the surfaces of set (15'), for such does not exist at any subsequent arbitrary time. Moreover, the point of phase agreement arises in suck a way ·that the sets (15) continuously exchange their representatives given to (15').
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That is shown thus. There must hold
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-(aw) -(aw) -(aw) (l6)
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w _ w aw
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aw
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aw
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- 0' Oa1 - oal o' aa2 - da2 0 • •• Oan - Oan o'
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simultaneously for the common meeting point of all members of (15')
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at any time, because the da/s are arbitrary within a small region. In these n + l equations, the right-hand sides are constants, and the left are functions of the n + 1 quantities q1, q2, ••• qn, t. The equations are satisfied by the initia] system of values, i.e. by the co-ordinates of P and the initial time t. For another arbitrary value oft, they will have no solutions in q1 . . . qn, but will more than define the system of these n quantities.
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We may proceed, however, as follows. Let us leave the first
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equation, W = W0, aside at first, and define the qk's as functions of
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the time and the constants according to the remaining n equations. Let this point be called Q. By it, naturally, the first equation will
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not be satisfied, but the left-hand side will differ from the right by a certain value. If we go back to the derivation of system (16) from (15'), what we have just said means that though Q is not a common point for the set of surfaces (15'), it is so, however, for a set which results from (15'), if we alter the right-hand side of equation (15') by an amount which is constant for all the surfaces. Let this new
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set be (15"). For it, therefore, Q is a common point. The new set results from (15'), as stated above, by an exchange of the representatives in (15'). This exchange is occasioned by the alteration of the constant in (15), by the sa,me amount, for all representatives. Hence the phase angle is altered by the same amount for all representatives. The new representatives, i.e. the members of the set we have called (15*), which meet· in the point Q, agree in phase angle just as
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the old ones did. This amounts therefore to saying:
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VV AV~ lVl~li.tlA.N llil.S
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The point Q which is defined as a function of the time by the n equations
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(17)
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continues to be a point of agreeing phase for the whole aggregate of
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wave sets (15).
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Of all the n-surfaces, of which Q is shown by (17) to be the common
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point, only the first is variable ; the others remain fixed (only the
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first of equations (17) contains the time). The n -1 fixed surfaces
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determine the path of the point Q as their line of intersection. It is
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easily shown that this line is the orthogonal trajectory of the set
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W =Const. For, by hypothesis, W satisfies the Hamilton-Jacobi equa-
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tion (l ') identically in a 1, a2 • • • an. H we now differentiate the
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aw Hamilton -Jacobi equation with respect to a1: (k =2, 3, . . . n),
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we get the statement that the normal to a surface, ~ =Const.,
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uak
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is perpendicul,ar, at every point on it, to the normal of the surface,
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W =Const., which passes through that point, i.e. that each of the two
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surfaces contains the normal to the other. If the line of intersection
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of the n -1 fixed surfaces (17) has no branches, as is generally the case,
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then must each line element of the intersection, as the sole common
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line element of the n - 1 surfaces, coincide with the normal of the
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W-surface, passing through the same point, i.e. the line of intersection
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is the orthogonal trajectory of the W-surfaces. Q.E.D.
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We may sum up the somewhat detailed discussion, which has led us
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to equations (17), in a much shorter or (so to speak) shorthand fashion,
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(i), as follows: W denotes, apart from a universal constant
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the
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phase angle of the wave function. If we now deal not merely with one, but with a continuous manifold of wave systems, and if these are continuously arranged by means of any continuous parameters
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% then the equations ~W =const. express the fact that all infinitely
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uai
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adjacent individuals (wave systems) of this manifold agree in phase. These equations therefore define the geometrical locus of the points of agreeing phase. If the equations are sufficient, this locus shrinks to one point; the equations then define the point of phase agreement as a function of the time.
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Since the system of equations (17) agrees with the known second system of equations of Jacobi, we have thus shown:
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The point of phase agreement for certain infinitesimal manifolds of wave systems, containing n parameters, moves according to the same l,aws
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as the image point of the meclianical system. I consider it a very difficult task to give an exact proof that the
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superposition of these wave systems really produces a noticeable disturbance in only a relatively small region surrounding the point of phase agreement, and that everywhere else they practically destroy
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|
QUANTISATION AND PROPER VALUES-II
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25
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one another through interference, or that the above statement turns out to be true at least for a suitable choice of the amplitudes, and possibly for a special choice of the form of the wave surfaces. I will advance the physical hypothesis, which I wish to attach to what is to be proved, without attempting the proof. The latter will only be worth while if the hypothesis stands the test of trial and if its application should require the exact proof.
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On the other hand, we may be sure that the region to which the disturbance may be confined still contains in all directions a great number of wave lengths. This is directly evident, firstly, because so Jong as we are only afew wave lengths distant from the point of phase agreement, then the agreement of phase is hardly disturbed, as the interference is still almost as favourable as it is at the point itself. Secondly, a glance at the three-dimensional Euclidean case of ordinary optics is sufficient to assure us of this general behaviour.
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What I now categorically conjecture is the following: The true mechanical process is realised or represented in a fitting way by the wave processes in q-space, and not by the motion of image points in this space. The study of the motion of image points, which is the object of classical mechanics, is only an approximate treatment, and has, as such, just as much justification as geometrical or " ray " optics has, compared with the true optical process. A macroscopic mechanical process will be portrayed as a wave signal of the kind described above, which can approximately enough be regarded as confined to a point compared with the geometrical structure of the path. We have seen that the same laws of motion hold exactly for such a signal or group of waves as are advanced by classical mechanics for the motion of the image point. This manner of treatment, however, loses all meaning where the structure of the path is no longer very large compared with the wave length or indeed is comparable with it. Then we must treat the matter strictly on the wave theory, i.e. we must proceed from the wave equation and not from the fundamental equations of mechanics, in order to form a picture of the manifold of the possible processes. These latter equations are just as useless for the elucidation of the micro-structure of mechanical processes as geometrical optics is for explaining the phenomena of diffraction. Now that a certain interpretation of this micro-structure has been successfully obtained as an addition to classical mechanics, although admittedly under new and very artificial assumptions, an interpretation bringing with it practical successes of the highest importance, it seems to me very significant that these theories-I refer to the forms of quantum theory favoured by Sommerfeld, Schwarzschild, Epstein, and others-bear a very close relation to the HamiltonJacobi equation and the theory of its solution, i.e. to that form of classical mechanics which already points out most clearly the true undulatory character of mechanical processes. The Hamilton-Jacobi equation corresponds to Huygens' Principle (in its old simple form, not in the form due to Kirchhoff). And just as this, supplemented by
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26
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|
|
WA VE MECHANICS
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|
some rules which are not intelligible in geometrical optics (Fresnel's construction of zones), can explain to a great extent the phenomena of diffraction, so light can be thrown on the processes in the atom by the theory of the action-function. But we inevitably became involved in irremovable contradictions if we tried, as was very natural, to maintain also the idea of paths of systems in these processes; just as we find the tracing of the course of a light ray to be meaningless, in the neighbourhood of a diffraction phenomenon.
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|
We can argue as follows. I will, however, not yet give a conclusive picture of the actual process, which positively cannot be arrived at from this starting-point but only from an investigation of the wave equation ; I will merely illustrate the matter qualitatively. Let us think of a wave group of the nature described above, which in some way gets into a small closed" path", whose dimensions are of the order of the wave length, and therefore small compared with the dimensions of the wave group itself. It is clear that then the " system path " in the sense of classical mechanics, i.e. the path of the point of exact phase agreement, will completely lose its prerogative, because there exists a whole continuum of points before, behind, and near the particular point, in which there is almost as complete phase agreement, and which describe totally different " paths ". In other words, the wave group not only fills the whole path domain all at once but also stretches far beyond it in all directions.
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|
In this sense do I interpret the "phase waves" which, according to de Broglie, accompany the path of the electron ; in the sense, therefore, that no special meaning is to be attached to the electronic path itself (at any rate, in the interior of the atom),and still less to the position of the electron on its path. And in this sense I explain the convict.ion, increasingly evident to-day, firstly, that real meaning has to be denied to the phase of electronic motions in the atom; secondly, that we can never assert that the electron at a definite instant is to be found on any definite one of the quantum paths, specialised by the quantum conditions ; and thirdly, that the true laws of quantum mechanics do not consist of definite rules for the single path, but that in these laws the elements of the whole manifold of paths of a system are bound together by equations, so that apparently a certain reciprocal action exists between the different paths.1
|
|
It is not incomprehensible that a careful analysis of the experimentally known quantities should lead to assertions of this kind, if the experimentally known facts are the outcome of such a structure of the real process as is here represented. All these assertions systematically contribute to the relinquishing of the ideas of "place of the electron" and " path of the electron ". If these are not given up, contradictions remain. This contradiction has been so strongly felt that it has even been doubted whether what goes on in the atom could ever be described within the scheme of space and time. From the philo-
|
|
1 Cf. especially the papers of Heisenberg, Born, Jordan, and Dirac quoted later, and further N. Bohr, Die NaturwiBBen11chaften, January 1926.
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|
QUANTISATION AND PROPER VALUES-II
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27
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sophical standpoint, I would consider a conclusive decision in this
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|
sense as equivalent to a complete surrender. For we cannot really
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|
alter our manner of thinking in space and time, and what we cannot
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|
comprehend within it we cannot understand at all. There are such
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|
things-but I do not believe that atomic structure is one of them.
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|
From our standpoint, however, there is no reason for such doubt,
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although or rather because its appearance is extraordinarily comprehen-
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sible. So might a person versed in geometrical optics, after many
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attempts to explain diffraction phenomena by means of the idea .of
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the ray (trustworthy for his macroscopic optics), which always came to
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nothing, at last think that the Laws of Geometry are not applicable to
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diffraction, since he continually finds that light rays, which he imagines
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as rectilinear and independent of each other, now suddenly show, even
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in homogeneous media, the most remarkable curvatures, and obviously
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mutually influence one another. I consider this analogy as very strict.
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Even for the unexplained curvatures, the analogy in the atom is not
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lacking-think of the "non-mechanical force", devised for the explana-
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tion of anomalous Zeeman effects.
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In what way now shall we have to proceed to the undulatory
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representation of mechanics for those cases where it is necessary 1
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We must start, not from the fundamental equations of mechanics, but
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from a wave equation for q-space and consider the manifold of processes
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possible according to it. The wave equation has not been explicitly
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used or even put forward in this communication. The only datum for
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its construction is the wave velocity, which is given by (6) or (6') as a
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|
|
function of the mechanical energy parameter or frequency respectively,
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|
and by this datum the wave equation is evidently not uniquely defined.
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It is not even decided that it must be definitely of the second order.
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Only the striving for simplicity leads us to try this to begin with.
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We will then say that for the wave function y, we have
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(18)
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div
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grad
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y,
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-
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1 u 2
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..
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ip
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=
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0,
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valid for all processes which only depend on the time through a factor e2rrivt. Therefore, considering (6), (6'), and (11), we get, respectively,
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87T2
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(18')
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div grad 1/1 + 7t2(hv - V)1/] =0,
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and
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(18")
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div
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grad
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ifJ
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81r2
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+ Ji2(E -
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V)1/I
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=0.
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|
The differential operations are to be understood with regard to the line element (3). But even under the postulation of second order, the above is not the only equation consistent with (6). For it is possible
|
|
to generalize by replacing div grad y, by
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|
(19)
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|
f(qk) div (/(~k) grad 1/1),
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|
where f may be an arbitrary function of the q's, which must depend in
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28
|
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|
|
WAVE MECHANICS
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|
some plausible way on E, V(qk), and the coefficients of the line element (3). (Think, e.g., off =u.) Our postulation is again dictated by the striving for simplicity, yet I consider in this case that a wrong deduction is not out of the question.1
|
|
The substitution of a partial differential equation for the equations of dynamics in atomic problems appears at first sight a very doubtful procedure, on account of the multitude of solutions that such an equation possesses. Already classical dynamics had led not just to one solution but to a much too extensive manifold of solutions, viz. to a continuous set, while all experience seems to show that only a discrete number of these solutions is realised. The problem of the quantum theory, according to prevailing conceptions, is to select by means of the "quantum conditions" that discrete set of actual paths out of the continuous set of paths possible according to classical mechanics. It seems to be a bad beginning for a new attempt in this direction if the number of possible solutions has been increased rather than diminished.
|
|
It is true that the problem of classical dynamics also allows itself to be presented in the form of a partial equation, namely, the HamiltonJacobi equation. But the manifold of solutions of the problem does not correspond to the manifold of solutions of that equation. An arbitrary " complete " solution of the equation solves the mechanical problem completely ; any other complete solution yields the same paths -they are only contained in another way in the manifold of paths.
|
|
Whatever the fear expressed about taking equation (18) as the foundation of atomic dynamics comes to, I will not positively assert that no further additional definitions will be required with it. But these will probably no longer be of such a completely strange and incomprehensible nature as the previous " quantum conditions ", but will be of the type that we are accustomed to find in physics with a partial differential equation as initial or boundary conditions. They will be, in no way, analogous to the quantum conditions-because in all cases of classical dynamics, which I have investigated up till now, it turns out that equation (18) carries within itself the quantum conditions. It distinguishes in certain cases, and indeed in those where experience demands it, of itself, certain frequencies or energy levels as those which alone are possible for stationary processes, without any further assumption, other than the almost obvious demand that, as a physical quantity, the function if, must be single-valued, finite, and continuous throughout configuration space.
|
|
Thus the fear expressed is transformed into its contrary, in any case in what concerns the energy levels, or let us say more prudently, the frequencies. (For the question of the "vibrational energy " stands by itself; we must not forget that it is only in the one electron problem that the interpretation as a vibration in real three-dimensional space is immediately suggested.) The definition of the quantum levels no
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1 The introduction of f(qk) means that not only the "density" but also the " elasticity " varies with the position.
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|
QUANTISATION AND PROPER VALUES-II
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29
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|
longer takes place in two separated stages: (1) Definition of all paths dynamically possible. (2)' Discarding of the greater part of those solutions and the selection of a few by special postulations ; on the contrary, the quantum levels are at once defined as the proper values of equation (18), which carries in itself its natural boundary conditions.
|
|
As to bow far an analytical simplification will be effected in this way in more complicated cases, I have not yet been able to decide. I should, however, expect so. Most of the analytical investigators have the feeling that in the two-stage process, described above, there must be yielded in (I) the solution of a more complicated problem than is really necessary for the final result: energy as a (usually) very simple rational function of the quantum numbers. Already, as is known, the applicationof theHamilton-Jacobi method creates a greatsimplification, as the actual calculation of the mechanical solution is avoided. It is sufficient to evaluate the integrals, which represent the momenta, merely for a closed complex path of integration instead of for a variable upper limit, and this gives much less trouble. Still the complete solution of the Hamilton-Jacobi equation must really be known, i.e. given by quadratures, so that the integration of the mechanical problem must in principle be effected for arbitrary initial values. •In seeking for the proper values of a differential equation, we must usually, in practice, proceed thus. We seek the solution, firstly, without regard to boundary or continuity conditions, and from the form of the solution then pick out those values of the parameters, for which the solution satisfies the given conditions. Part I. supplies an example of this. We see by this example also, however-what is typical of proper value problems-that the solution was only given generally in an extremely inaccessible analytical form [equation (12) loc. cit.], but that it is extraordinarily simplified for those proper values belonging to the " natural boundary condition ". I am not well enough informed to say whether direct methods have now been worked out for the calculation of the proper values. This is known to be so for the distribution of proper values of high order. But this limiting case is not of interest here ; it corresponds to the classical, macroscopic mechanics. For spectroscopy and atomic physics, in general just the first 5 or IO proper values will be of interest; even the first alone would be a great result-it defines the ionisation potential. From the idea, definitely outlined, that every problem of proper values allows itself to be treated as one of maxima and minima without direct reference to the differential equation, it appears to me very probable that direct methods will be found for the calculation, at least approximately, of the proper values, as soon as urgent need arises. At' least it should be possible to test in individual cases whether the proper values, known numerically to all desired accuracy through spectroscopy, satisfy the problem or not.
|
|
I would not like to proceed without mentioning here that at the present time a research is being prosecuted by Heisenberg, Born,
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30
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WAVE MECHANICS
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Jordan, and other distinguished workers,1 to remove the quantum difficulties, which has already yielded such noteworthy success that it cannot be doubted that it contains at least a part of the truth. In its tendency, Heis_enberg's attempt stands very near the present one, as we have already mentioned. In its method, it is so totally different that I have not yet succeeded in finding the connecting link. I am distinctly hopeful that these two advances will not fight against one another, but on the contrary, just because of the extraordinary difference between the starting-points and between the methods, that they will supplement one another and that the one will make progress where the other fails. The strength of Heisenberg's programme lies in the fact that it promises to give the line-intensities, a question that we have not approached as yet. The strength of the present attempt-if I may be permitted to pronounce thereon-lies in the guiding, physical point of view, which creates a bridge between the macroscopic and microscopic mechanical processes, and which makes intelligible the outwardly different modes of treatment which they demand. For me, personally, there is a special charm in the conception, mentioned at the end of the previous part, of the emitted frequencies as " beats ", which I believe will lead to an intuitive understanding of the intensity formulae.
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§ 3. Application to Examples
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We will now add a few more examples to the Kepler problem treated in Part I., but they will only be of the very simplest nature, since we have provisionally confined ourselves to classical mechanics, with no magnetic field. 2
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I. The Planck Osc1:llator. The Question of Degeneracy
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Firstly we will consider the one-dimensional oscillator. Let the
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co-ordinate q be the displacement multiplied by the square root of
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the mass. The two forms of the kinetic energy then are
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(20)
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T = ½<P, T =½p2•
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The potential energy will be
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(21)
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V(q) =2772v02q2,
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where v0 is the proper frequency in the mechanical sense. Then equation (18) reads in this case
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(22)
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dd2qi2fi +78,1,2r2(E
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-
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2 2 1T Vo
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2q2).·.'·,,-,-
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0
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•
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1 W. Heisenberg, Ztschr. J. Phys. 33, p. 879, 1925; M. Born and P. Jordan, ibid. 34,
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p. 858, 1925; M. Bom, W. Heisenberg, and P. Jordan, ibid. 35, p. 557, 1926; P. Dirac,
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Proc. Roy. Soc., London, 109, p. 642, 1925. 2 In relativity mechanics and taking a magnetic field into account the statement
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of the Hamilton-Jacobi equation becomes more complicated. In the case of a single electron, it asserts that the four-dimensional gradient of the action function, diminished
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by a given vector (the four-potential), has a constant value. The translation of this statement into the language of the wave theory presents a good many difficulties.
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QUANTISATION AND PROP.KH. VALU~S-11
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31
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For brevity write (23) Therefore (22')
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81r2E a=~'
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Introduce as independent variable
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(24)
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X = qvb,
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and obtain
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The proper values and functions of this equation are known. 1 The proper values are, with the notation used here,
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(25)
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va'b = 1, 3, 5 ... (2n + 1) ...
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The functions are the orthogonal functions of Hermite,
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(26)
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e- 2rfn(x).
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Hn(x) means the nth Hermite polynomial, which can be defined as
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drte-zl
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(27)
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Hn(x) = ( - l)nez2 dxn ,
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or explicitly by
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(27')
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Hn(x)
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=
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(2xr-
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n(nl
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-1) ! (2x)"-
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2
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n(n +
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l)(n 2!
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2)(n -
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3)(2X)n- 4 -+.
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The first of these polynomials are
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(27")
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H0(x) = 1
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H1(x) =2x
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H 2(x) = 4x2 - 2
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Ha(x) =8x3 -12x
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Hix)= 16x4 - 48x2 + 12
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Considering next the proper values, we get from (25) and (23)
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(25')
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En=2n-2+- l hv0 ; n=O, 1, 2, 3, . . .
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Thus as quantum levels appear so-called " half-integral " multiples of the "quantum of energy" peculiar to the oscillator, i.e. the odd
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multiples of h;o. The intervals between the levels, which alone are
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important for the radiation, are the same as in the former theory. It is remarkable that our quantum levels are exactly those of Heisenberg's theory. In the theory of specific heat this deviation from the previous
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1 Cf. Coura.nt-Hilbert, Methoda of Mathematical Physics, i. (Berlin, Springor, 1924), v. § 9, p. 261, eqn. 43, and further ii. § 10, 4, p. 76.
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32
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WAVE MECHANICS
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theory is not without significance. It becomes important first when
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the proper frequency v0 varies owing to the dissipation of heat. Formally it has to do with the old question of the " zero-point energy ",
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which was raised in connection with the choice between the first
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and second forms of Planck's Theory. By the way, the additional
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-1 term hv also influences the law of the b· and-e,dges.
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The proper funct,ions (26) become, if we reintroduce the original
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q from (24) and (23),
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(26')
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). Y,n(q) = e- 2rrj,oq'Hn(27Tq✓~
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Consideration of (27") shows that the first function is a Gaussian Error-curve; the second vanishes at the origin and for x positive corresponds to a "Maxwell distribution of velocities" in two dimensions, and is continued in the manner of an odd function for x negative. The third function is even, is negative at the origin, and has two
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symmetrical zeros at ± ~ ' etc. The curves can easily be sketched
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roughly and it is seen that the roots of consecutive polynomials separate one another. From (26') it is also seen that the characteristic points of the proper functions, such as half-breadth (for n = 0), zeros, and maxima, are, as regards order of magnitude, within the range of the classical vibration of the oscillator. For the classical amplitude of the nth vibration is readily found to be given by
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(28)
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qn
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=
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'\f'En
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27TVo
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=
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I
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27T
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\ /(~h
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'V{~ 2n+'f
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Yet there is in general, as far as I see, no definite meaning that can be attached to the exact abscissa of the classical turning points in the graph of the proper function. It may, however, be conjectured, because the turning points have this significance for the phase space wave, that, at them, the square of the velocity of propagation becomes in.finite and at greater distances becomes negative. In the differential equation (22), however, this only means the vanishing of the coefficient of if, and gives rise to no singularities.
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I would not like to suppress the remark here (and it is valid quite generally, not merely for the oscillator), that nevertheless this vanishing and becoming imaginary of the velocity of propagation is something which is very characteristic. It is the analytical reason for the selection of definite proper values, merely through the condition that the function should remain finite. I would like to illustrate this further. A wave equation with a real velocity of propagation means just this : there is an accelerated increase in the value of the function at all those points where its value is lower than the average of the values at neighbouring points, and vice versa. Such an equation, if not immediately and lastingly as in case of the equation for the conduction of heat, yet in the course of time, causes a levelling
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~UA.NT1~AT1U1'1 Al~.lJ r.ttvr.r...tt VALUJ!u~-.1.1
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.).)
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of extreme values and does not permit at any point an excessive growth of the function. A wave equation with an imaginary velocity of propagation means the exact opposite : values of the function above the average of surrounding values experience an accelerated increase (or retarded decrease), and vice versa. We see, therefore, that a function represented by such an equation is in the greatest danger of growing beyond all bounds, and we must order matters skilfully to preserve it from this danger. The sharply defined proper values are just what makes this possible. Indeed, we can see in the example treated in Part I. that the demand for sharply defined proper values immediately ceases as soon as we choose the quantity E to be positive, as this makes the wave velocity real throughout all space.
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After this digression, let us return to the oscillator and ask ourselves if anything is altered when we allow it two or more degrees of freedom (space oscillator, rigid body). If different mechanical proper frequencies (v0-values) belong to the separate co-ordinates, then nothing is changed. if, is taken as the product of functions, each of a single co-ordinate, and the problem splits up into just as many separate problems of the type treated above as there are co-ordinates present. The proper functions are products of Hermite orthogonal functions, and the proper values of the whole problem appear as sums of those of the separate problems, taken in every possible combination. No proper value (for the whole system) is multiple, if we presume that there is no rational relation between the v0-values.
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If, however, there is such a relation, then the same manner of treatment is still possible, but it is certainly not unique. Multiple proper values appear and the "separation" can certainly be effected in other co-ordinates, e.g. in the case of the isotropic space oscillator in spherical polars.1
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The proper values that we get, however, are certainly in each case exactly the same, at least in so far as we are able to prove the " completeness " of a system of proper functions, obtained in one way. We recognise here a complete parallel to the well-known relations which the method of the previous quantisation meets with in the case of de,generacy. Only in one point there is a not unwelcome formal difference. If we applied the Sommerfeld-Epstein quantum conditions without regard to a possible degeneracy then we always got the same energy levels, but reached different conclusions as to the paths permitted, according to the choice of co-ordinates.
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Now that is not the case here. Indeed we come to a completely different system of proper functions, if we, for example, treat the vibration problem corresponding to unperturbed Kepler motion in
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1 We are led thus to an equation in r, which may be treated by the method shown in the Kepler problem of Part I. Moreover, the one-dimensional oscillator leads to the same equation if q2 be taken as variable. I originally solved the problem directly in that way. For the hint that it was a question of Hermite polynomials, I have to thank Herr E. Fues. The polynomial appearing in the Kepler problem (eqn. 18 of Part I.) is the (2n+l)th differential coefficient of the (n+l)th polynomial of Laguerre, as I subsequently found.
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34
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WAVE M.h:UHA.NlU~
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parabolic co-ordinates instead of the polars used in Part I. However, it is not just the single proper vibration that furnishes a possible state of vibration, but an arbitrary, finite or infinite, linear aggregate of such vibrations. And as such the proper functions found in any second way may always be represented; namely, they may be represented as linear aggregates of the proper functions found in an arbitrary way, provided the latter form a complete system.
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The question of how the energy is really distributed among the proper vibrations, which has not been taken into account here up till now, will, of course, have to be faced some time. Relying on the former quantum theory, we will be disposed to assume that in the degenerate case only the energy of the set of vibrations belonging to one definite proper value must have a certain prescribed value, which in the non-degenerate case belongs to one single proper vibration. I would like to leave this question still quite open-and also the question whether the discovered " energy levels " are really energy steps of the vibration process or whether they merely have the significance of its frequency. If we accept the beat theory, then the meaning of energy levels is no longer necessary for the explanation of sharp emission frequencies.
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2. Rotator with Fixed Axis
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|
On account of the lack of potential energy and because of the
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Euclidean line element, this is the simplest conceivable example of
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vibration theory. Let A be the moment of inertia and <p the angle
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of rotation, then we clearly obtain as the vibration equation
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|
1 d2ip 81r2E
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|
(29)
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A d</>2 + ~ i f , =0,
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|
which has the solution
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(30)
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if,=
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sin cos
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[
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'V{8~ ;2EA..
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<f>].
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Here the argument must be an integral multiple of <p, simply because otherwise if, would neither be single-valued nor continuous throughout the range of the co-ordinate <p, as we know <p + 21r has the same significance as <p. This condition gives the. well-known result
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|
n2h2
|
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|
(31)
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|
|
En= B1r2A
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|
in complete agreement with the former quantisation. No meaning, however, can be attached to the result of the application
|
|
to band spectra. For, as we shall learn in a moment, it is a peculiar fact that our theory gives another result for the rotator with free axis. And this is true in general. It is not allowable in the applications of wave mechanics, to think of the freedom of movement of the system as being more strictly limited, in order to simplify calculation, than it actually is, even when we know from the integrals of the mechanical
|
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|
"(,V~.L.1.~ .&..&...._,..__._..&..a,'-'.._~ .... _._..._,_ .a._.....,.....,-._...__...,. '..,,._ _ _ _ .....,,,
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|
|
--
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|
equations that in a single movement certain definite freedoms are not made use of. For micro-mechanics, the fundamental system of mechanical equations is absolutely incompetent; the single paths with which it deals have now no separate existence. A wave process fills the whole of the phase space. It is well known that even the number of the dimensions in which a wave process takes place is very significant.
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3. Rigid Rotator with Free Axis
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|
|
If we introduce as co-ordinates the polar angles 0, <p of the radius
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|
|
from the nucleus, then for the kinetic energy as a function of the
|
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|
|
momenta we get (32)
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|
K ) T = 2lA (PrJ2 + sin2 0 •
|
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|
According to its form this is the kinetic energy of a particle constrained
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|
|
to move on a spherical surface. The Laplacian operator is thus simply
|
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|
|
that part of the spatial Laplacian operator which depends on the polar
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|
|
angles, and the vibration equation (18") takes the following form,
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|
(33)
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1 sin
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0
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a
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30
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( .
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sm
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|
8aaeif,)
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+
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1 sin 2
|
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0
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|
|
oo<2fi,f2,
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+
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|
81r 2A E
|
|
~ip=
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O
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|
|
.
|
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|
|
The postulation that ip should be single-valued and continuous on the spherical surface leads to the proper value condition
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|
(34)
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|
81r2A liTE
|
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=n(n
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+
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1);
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|
n =0, l, 2, 3, ...
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|
The proper functions are known to be spherical surface harmonics. The energy levels are, therefore,
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|
|
n(n + l}h2
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|
(34')
|
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|
|
En= B1r2A ; n =0, l, 2, 3,
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|
This definition is different from all previous statements (except perhaps that of Heisenberg 1). Yet, from various arguments from experiment we were led to put " half-integral " values for n in formula (31). It is easily seen that (34') gives practically the same as (31) with half-integral values of n. For
|
|
n(n + 1) = (n + ½) 2 - ¼-
|
|
The discrepancy consists only of a small additive constant; the level differences in (34') are the same as are got from "half-integral quantisation". This is true also for the application to short-wave bands, where the moment of inertia is not the same in the initial and final states, on account of the " electronic jump ". For at most a small constant additional part comes in for all lines of a band, which is swamped in the large "electronic term " or in the "nuclear vibration term". Moreover, our previous analysis does not permit us to speak of this small part in any more definite way than as, say,
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4 1
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|
h2 (
|
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81r2
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A1
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|
-
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1 A'
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)
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•
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|
n ..t"l. t' .1.!.I .l.U.l.!.lv.1..1..n...i.., .1.v..._,
|
|
The notion of the moment of inertia being fixed by "quantum conditions " for electronic motions and nuclear vibrations follows naturally from the whole line of thought developed here. We will show in the next. section how we can treat, approximately at least, the nuclear vibrations and the rotations of the diatomic molecule simultaneously by a synthesis of the cases 1 considered in 1 and 3.
|
|
I should like to mention also that the value n = 0 corresponds not
|
|
to the vanishing of the wave function ip but to a constant value for
|
|
it, and accordingly to a vibration with amplitude constant over the whole sphere.
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|
|
4. Non-rigid Rotator (Diatomic Molecule)
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|
According to the observation at the end of section 2, we must state the problem initially with all the six degrees of freedom that the rotator really possesses. Choose Cartesian co-ordinates for the two molecules, viz. x 1, y1, z1 ; x 2, y2, z 2, and let the masses be m 1 and m2, and r be their distance apart. The potential energy is
|
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|
|
(35)
|
|
where Here
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|
|
V = 21r2v02µ,(r - r0) 2, r 2 = (x1 - x2) 2 + (y1 -y2) 2 + (z1 - z2) 2•
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|
|
(36)
|
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|
|
may be called the "resultant mass". Then v0 is the mechanical proper frequency of the nuclear vibration, regarding the line joining
|
|
the nuclei as fixed, and r0 is the distance apart for which the potential energy is a minimum. These definitions are all in the sense
|
|
of the usual mechanics.
|
|
For the vibration equation (18") we get the following :
|
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|
|
1 ( a2ifi a2ip a2ifi) 1 ( a2ifi a2ifi a2ip)
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|
oyl (37) ml OX12 + OY12 + OZ12 + m2 OX22 +
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|
+ OZ22
|
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1
|
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|
|
81r2
|
|
+11[E -
|
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|
|
21T2v02µ,(r
|
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|
|
-
|
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|
r0)2]
|
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|
|
ip = 0.
|
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|
|
Introduce new independent variables x, y, z, g, TJ, ,, where
|
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|
|
(38)
|
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|
|
+ X = X1 - X2; (m1 m2)f =m1X1 + m2X2
|
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|
|
Y =Y1 -Y2; (m1 + m2)1J =m1Y1 + miJh
|
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|
|
z + =z1 -z2 ; (m1 m 2)' =m1z 1 +m2z 2.
|
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|
|
The substitution gives
|
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|
|
v,) v,) (a a a (a a a µ,1 2ifi 2i/J 2
|
|
|
|
1 2ifi 2ifi 2
|
|
|
|
(37') { oy OX2 + 2 + OZ2 + ml +m2 of2 + 01) 2 + 0,2
|
|
|
|
+ [a" -b'(r -r0) 2] l/, = 0,
|
|
|
|
where for brevity
|
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|
|
1 Cf. A: Sommerfeld, Atombau und Spektrallinien, 4th edit., p. 833. We do not consider here the additional non-harmonic terms in the potential energy.
|
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|
|
(39)
|
|
|
|
" 81r2E b' = l61r4vo2fL
|
|
|
|
a=~,
|
|
|
|
h2 •
|
|
|
|
Now we can put for 4, the product of a function of the relative coordinates x, y, z, and a function of the co-ordinates of the centre of mass
|
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|
|
[, 71, ,:
|
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|
|
(40)
|
|
|
|
if, =f(x, y, z) g ([, 71, ().
|
|
|
|
For g we get the defining equation
|
|
|
|
(41)
|
|
|
|
(0 ml +1m2
|
|
|
|
29 029 029 of2+0712+ol2
|
|
|
|
)
|
|
|
|
+const.
|
|
|
|
g=O.
|
|
|
|
This is of the same form as the equation for the motion, under no
|
|
|
|
forces, of a particle of mass m1 + m2. The constant would in this case have the meaning
|
|
|
|
(42)
|
|
|
|
const. = 8h1T22Et'
|
|
|
|
where Et is the energy of translation of the said particle. Imagine this value inserted in (41). The question as to the values of Et admissible as proper values depends now on this, whether the whole infinite space is available for the original co-ordinates and hence for those of the centre of gravity without new potential energies coming in, or not. In the first ·case every non-negative value is permissible and everynegativevalue not permissible. For when Et is not negative and only then, (41) possesses solutions which do not vanish identically and yet remain finite in all space. If, however, the molecule is situated in a " vessel ", then the latter must supply boundary conditions for the function g, or in other words, equation (41), on account of the introduction of further potential energies, will alter its form very abruptly at the walls of the vessel, and thus a discrete set of Ecvalues will be selected as proper values. It is a question of the " Quantisation of the motion of translation ", the main points of which I have lately discussed, showing that it leads to Einstein's Gas Theory.1
|
|
For the factorf of the vibration function 4,, depending on the relative co-ordinates x, y, z, we get the defining equation
|
|
|
|
(43)
|
|
|
|
µ.1 (0o2x.f2+o02y.2f +ooz22f) +[aI -bI (r-ro)2]f=0,
|
|
|
|
where for brevity we put
|
|
|
|
(39')
|
|
|
|
We now introduce instead of x, y, z, the spherical polars r, 0, cp (which is
|
|
in agreement with the previous use of r). After multiplying byµ, we get
|
|
|
|
f) ii) (43')
|
|
|
|
r!2o~r(r2o8r
|
|
|
|
+ lr_2.lfs_inle
|
|
|
|
~-(sin 00
|
|
|
|
00
|
|
|
|
+ s_inl2_0 a042>12}
|
|
|
|
+ [µ.a' -µ.b'(r - r0)2]f = 0.
|
|
|
|
1 Physik. Ztschr. 27, p. 95, I 926.
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|
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38
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WA V..l:!i 1Vl~lil1.Al'41L,i:,
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Now break up f. The factor depending on the angles is a surface harmonic. Let the order be n. The curled bracket is - n(n + l )f. Imagine this inserted and for simplicity let f now stand for the factor depending on r. Then introduce as new dependent variable
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(44)
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X =rf,
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and as new independent variable
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(45)
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p =r -r0•
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The substitution gives
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(46)
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a2x
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ap2+
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[
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µ
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,
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,
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a
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-
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µ
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,
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b'
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p 2
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-
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n(n+l) (ro+p)2
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] x
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-_O
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.
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To this point the analysis has been exact. Now we will make an approximation, which I well know requires a stricter justification than I will give here. Compare (46) with equation (22') treated earlier. They agree in form and only differ in the coefficient of the unknown
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function by terms of the relative order of magnitude of !!. . This is seen,
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To
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if we develop thus :
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(47 )
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n(n + 1) (ro + p) 2
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_
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-
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n(n + ro 2
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1)(1
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_
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2p ro
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+
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3p2_ ro 2
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+
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'
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•
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'
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) '
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substitute in (46), and arrange in powers of p/r0. If we introduce for p a new variable differing only by a small constant, viz.
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(48 )
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,
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n(n+l)
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a( p =p-
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'
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1o
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b'
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J-L
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+
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3n(n + ro 4
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1))
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then equation (46) takes the form
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(46')
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;}; + (a - bp'2 + [~])x = o,
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where we have put
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(49)
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J , a=µ,a
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-
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n(n + r2
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0
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l
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)(1
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n(n + l) - r04µ , b ' + 3 n ( n + l )
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)
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lb=µ/,'+ 3n(;.: I)_
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The symbol[~] in (46') represents terms which are small compared with
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the retained term of the order of i_.
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ro Now we know that the first proper functions of equation (22'), to which we now compare (46'), only differ markedly from zero in a small range on both sides of the origin. Only those of higher order stretch gradually further out. For moderate orders, the domain
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[i] for equation (46'), if we neglect the term and bear in mind the ro.
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(.lUAl~Tli::,A'.llUl~ Al'UJ r .n,vr.n,.n, V11.LU.Co:,-u
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,J,7
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order of magnitude of molecular constants, is indeed small compared
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with r0. We thus conclude (without rigorous proof, I repeat), that we can in this way obtain a useful approximation for the first proper functions, within the region where they differ at all markedly from zero, and also for the first proper values. From the proper value condition (25) and omitting the abbreviations (49), (39'), and (39), though introducing the small quantity
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(50)
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E = -n'-(-n--+--l)-h'-2- = n-(-n-'+--l)-h-2
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167r4vo2µ,2ro4 l6114vo2A2
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instead, we can easily derive the following energy steps,
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n(n+l)h2(
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e ) 2l+l . ~
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(51)
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E:E,+ {
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B1r•A . 1 =l + 3• + -2 -hv0v I +3e
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(n-0, I, 2 . . . , l-0, 1, 2 . . .),
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where
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(52)
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A= µ,r02
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is still written for the moment of inertia. In the language of classical mechanics, e: is the square of the ratio
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of the frequency of rotation to the vibration frequency v0 ; it is therefore really a small quantity in the application to the molecule, and formula (51) has the usual structure, apart from this small correction and the other differences already mentioned. It is the synthesis of (25') and (34') to which Et is added as representing the energy of translation. It must be emphasized that the value of the approximation is to be judged not only by the smallness of e: but also by l not being too large. Practically, however, only small numbers have to be considered for l.
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The e:-corrections in (51) do not yet take account of deviations of the nuclear vibrations from the pure harmonic type. Thus a comparison with Kratzer's formula (vide Sommerfeld, loc. cit.) and with experience is impossible. I only desired to mention the case provisionally, as an example showing that the intuitive idea of the equilibrium configuration of the nuclear system retains its meaning in undulatory mechanics also, and showing the manner in which it does so, provided that the wave amplitude if, is different from zero practically only in a small neighbourhood of the equilibrium configuration. The direct interpretation of this wave function of six variables in three-dimensional space meets, at any rate initially, with difficulties of an abstract nature.
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The rotation-vibration-problem of the diatomic molecule will have to be re-attacked presently, the non-harmonic terms in the energy of binding being taken into account. The method, selected skilfully by Kratzer for the classical mechanical treatment, is also suitable for undulatory mechanics. If, however, we are going to push the calculation as far as is necessary for the fineness of band structure, then we must make use of the theory of the perturbation of proper values and functions, that is, of the alteration experienced by a definite proper value and the appertaining proper functions of a
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40
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WA V..I!: M.l!JUHA.N lUS
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differential equation, when there is added to the coefficient of the unknown function in the equation a small "disturbing term". This "perturbation theory" is the complete counterpart of that of classical mechanics, except that it is simpler because in undulatory mechanics we are always in the domain of linear relations. As a first approximation we have the statement that the perturbation of the proper value is equal to the perturbing term averaged" over the undisturbed motion''.
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The perturbation theory broadens the analytical range of the new theory extraordinarily. As an important practical success, let me say here that the St,ark effect of the first order will be found to be really completely in accord with Epstein's formula, which has become unimpeachable through the confirmation of experience.
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Zurich, Physical Institute of the University. (Received February 23, 1926.)
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The Continuous Transition from Microto Macro-Mechanics
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(Die NaJ,urwissenschaften, 28, pp. 664-666, 1926)
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BUILDING on ideas of de Broglie 1 and Einstein,2 I have tried to show 3 that the usual differential equations of mechanics, which attempt to define the co-ordinates of a mechanical system as functions of the time, are no longer applicable for "small" systems; instead there must Qe introduced a certain partial differential equation, which defines a variable if, (" wave function") as a function of the coordinates and the time. As in the differential equation of a vibrating string or of any other vibrating system, if, is given as a superposition of pure time harmonic (i.e. "sinusoidal") vibrations, the frequencies of which agree exactly with the spectroscopic " term frequencies " of the micro-mechanical system. For example, in the case of the linear Planck oscillator 4 where the energy function is
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(1)
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2m(dq)2
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dt,
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+
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2 2 2-»,,r,2
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1T Vo ""'J. '
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when we put, instead of the displacement q, the dimensionless variable
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(2)
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x=q. 21rJm;~,
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we get if, as the superposition of the following proper vibrations : 5
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-f Y,n = e H n(x)e21rivnt
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(3)
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{ (
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v
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n
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2n+l =-2- v0
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;
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n=O, 1, 2, 3 . . . ).
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The Hn's are the polynomials 6 named after Hermite. If they are
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1 L. de Broglie, Ann. de Physique (10), 3, p. 22, 1925 (Theses, Paris, 1924). 2 A. Einstein, Berlin Ber. 1925, p. 9 et seq. 8 Ann. d. Physik; the essay!'! here collected. ' i.e. a particle of mass m which, moving in a straight line, is attracted towards a fixed point in it, with a force proportional to its displacement q from this point ; according to the usual mechanics, such a particle executes sine vibrations of frequency 110• 5 •means~- On the right-hand side the real part is to be taken, as usual. 8 Cf. Courant-Hilbert, Methoden der mathematischen Physik, I. chap. ii. § 10, 4, p. 76 (Berlin, Springer, 1924).
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41
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42
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WAVE MECHANICS
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x•
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multiplied by e-2 and the "normalising factor" (2nn !)-½ they are
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|
called Hermite's orthogonal functions. They represent therefore the amplitudes of the proper vibrations.
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The first five are represented in Fig. 1. The similarity between this and the well-known picture of the vibrations of a string is very great.
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At first sight it appears very strange to try to describe a process, which we previously regarded as belonging to particle mechanics, by a system of such proper vibrations. For this chosen simple case, I would like to demonstrate here in concreto the transition to macroscopic mechanics, by showing that a group of proper vibrations of high order-number n (" quantum number") and of relatively small order-number differences (" quantum number differences") may represent a "particle", which is executing the "motion", expected
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Fro. 1.-The first five proper vibrations of the Planck oscillator according to undulatory
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|
mechanics. Outside of the region - 3 :s;; x.:s;; + 3 represented here, all five functions
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approach the x-axis in monotonic fashion.
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from the usual mechanics, i.e. oscillating with the frequency v0. I choose a number A» l (i.e. great compared with 1) and form the following aggregate of proper vibrations:
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(4)
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'P = i ~ (~)n'Pn =eTTi11ot
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(1e21ri11ot)n _!__e-f Hn(x).
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n=O 2- n !
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n=O 2
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n !
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Thus the normalised proper vibrations (see above) are taken with the coefficients
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(5)
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and this, as is easily seen,1 results in the singling out of a relatively small group in the neighbourhood of the n-value given by
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A2
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(6)
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n = 2- ·
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1 zn/n ! has, as function of n,for large values of z, a single extremely high and relatively
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|
very sharp maximum at n=z. By taking square roots and with z=A 2/2, we get the series of numbers (5).
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MICRO- AND MACRO-M~CHANlCS
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The summation of the series (4) is made possible by the following
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identity 1 in x and s :
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(7)
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--n ( ) k~~J
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n=O
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n
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S
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1n
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e •
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~
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2
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n x
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=e
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-
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s
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1
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+
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2
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s
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z
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~
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-2-.
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Thus
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(8)
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Now we take, as is provided for, the real part of the right-hand side and after a short calculation obtain
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(9) if,= e~•-t(x-.A cos 2"v0t)2 cos [ 1rv0t + (A sin 21rv0t). ( X - : cos 21rv0t)].
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This is the final result, in which the first factor is our first interest. It represents a relatively tall and narrow " hump ", of the form of a " Gaussian error-curve ". which at a given moment lies in the neighbourhood of the position
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(10)
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x = A cos 21rv0t.
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The breadth of the hump is of the order of magnitude unity and
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therefore very small compared with A, by hypothesis. According to
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(.10), the hump oscillates under exactly the same law as would operate
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in the usual mechanics for a particle having (1) as its energy function.
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The amplitude in terms of xis A, and thus in terms of q is
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(11)
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a=A11i..f h.
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21r" mv0
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Ordinary mechanics gives for the energy of a particle of mass m, which
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oscillates with this amplitude and with frequency v0,
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(12)
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i.e. from (6) exactly nhv0, where n is the average quantum number of the selected group. The " correspondence " is thus complete in this respect also.
|
|
The second factor in (9) is in general a function whose absolute value is small compared with unity, and which varies very rapidly with x and also t. It ploughs many deep and narrow furrows in the profile of the first factor, and makes a wave group out of it, which is represented-schematically only-in Fig. 2. The x-scale of Fig. 2 is naturally much smaller than that of Fig. 1; Fig. 2 requires to be magnified five times before being directly compared with Fig. I. A more exact consideration of the second factor of (9) discloses the following interesting details, which cannot be seen in Fig. 2, which only represents one stage. The number and lYreadth of the "furrows" or "wavelets" within the particle vary with the time. The wavelets are most numerous and narrowest when passing through the centre x = 0 ; they become completely smoothed out at the turning points x =±A, because
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1 Courant-Hilbert, loc. cit. eqn. (58).
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44
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WAVE MECHANICS
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|
there, by (10), cos 2TTv0t = ± I and thus sin 2TTv0t becomes equal to zero, so that the second factor of (9) is absolutely independent of x. The entire extension of the wave group(" density of the particle") remains, however, always the same. The variability of the "corrugation" is to be conceived as depending on the velocity, and, as such, is completely intelligible from all general aspects of undulatory mechanics-but I do not wish to discuss this further at present.
|
|
Our wave group always remains compact, and does not spread out into larger regions as time goes on, as we were accustomed to make it do, for example, in optics. It is admitted that this does not mean much in one dimension, and that a hump on a string will behave quite similarly. But it is easily seen that, by multiplying together two or three expressions like (4), written in x, in y, and in z respectively, we can represent also the plane and the spatial oscillator respectively, i.e. a plane or spatial wave group which moves round a harmonic ellipse.1 Also such a wave group will remain compact, in contrast, e.g., to a
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|
-x~_~m------_-s_ _ _ _o_ _ _+_s_ _ _ _+_1i~v~
|
|
F10. 2.-0scillating wave group as the representation of a particle in wave mechanics.
|
|
wave packet in classical optics, which is dissipated in the course of time. The distinction may originate in the fact that our gro'..lp is built up out of separate discrete harmonic components, and not out of a continuum of such.
|
|
I wish to mention, in conclusion, that a general additive constant, O,let us say, which should be added to all the vn's in (3), (and corresponds to the "rest-energy" of the particle) does not alter the essentials. It only affects the square bracket in (9), adding 2TTCt thereto. Hence the oscillationR within the wave group become very much quicker with respect to the time, while the oscillation of the group as a whole, given by (10), and its "corrugation", remain quite unaffected.
|
|
We can definitely foresee that, in a similar way, wave groups can be constructed which move round highly quantised Kepler ellipses and are the representation by wave mechanics of the hydrogen electron. But the technical difficulties in the calculation are greater than in the especially simple case which we have treated here.
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|
1 We may point out, in passing, the interesting fact that the quantum levels of the plane oscillator are integral, but for the spatial oscillator they again become "halfintegral ". Similarly for the rotator. This ha.lf-integralness, which is spectroscopically so significant, is thus connected with the" oddness" of the number of the dimensions of space.
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On the Relation between the Quantum
|
|
Mechanics of Heisenberg, Born, and
|
|
Jordan, and that of Schrodinger
|
|
(Annalen der Physik (4), vol. 79, 1926)
|
|
§ I. Introduction and Abstract
|
|
CONSIDERING the extraordinary differences between the starting-points and the concepts of Heisenberg's quantum mechanics 1 and of the theory which has been designated " undulatory " or " physical " mechanics, 2 and has lately been described here, it is very strange that these two new theories agree with one another with regard to the known facts, where they differ from the old quantum theory. I refer, in particular, to the peculiar "half-integralness " which arises in connection with the oscillator and the rotator. That is really very remarkable, because starting-points, presentations, methods, and in fact the whole mathematical apparatus, seem fundamentally different. Above all, however, the departure from classical mechanics in the two theories seems to occur in diametrically opposed directions. In Heisenberg's work the classical continuous variables are replaced by systems of discrete numerical quantities (matrices), which depend on a pair of integral indices, and are defined by algebraic equations. The authors themselves describe the theory as a "true theory of a discontinuum ".3 On the other hand, wave mechanics shows just the reverse tendency; it is a step from classical point-mechanics towards a continuum-theory. In place of a process described in terms of a finite number of dependent variables occurring in a finite number of total differential equations, we have a continuous .fie,ld-like process in
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|
1 W. Heisenberg, Ztschr.f. Phys. 33, p. 879, 1925; M. Born and P. Jordan, idem 34, p. 858, 1925, and 35, p. 557, 1926 (the latter in collaboration with Heisenberg). I may be allowed, for brevity's sake, to replace the three names simply by Heisenberg, and to quote the ]ast two essays as" Quantum Mechanics I. and II." Interesting contributions to the theory have also been made by P. Dirac, Proc. Roy. Soc., London, 109, p. 642, 1925, and idem 110, p. 561, 1926.
|
|
2 E. Schrodinger. Parts I. and II. in this collection. These parts will be continued quite independently of the present paper, which is only intended to serve as a connecting link.
|
|
3 " Quantum Mechanics I." p. 879. 45
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46
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WA V.h: M~CHAl'HC8
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|
configuration space, which is governed by a single partial differential equation, derived from a principle of action. This principle and this differential equation replace the equations of motion and the quantum conditions of the older " classical quantum theory ".1
|
|
In what follows the very intimate inner connection between. Heisenberg's quantum mechanics and my wave mechanics will be disclosed. From the formal mathematical standpoint, one might well speak of the identity of the two theories. The train of thought in the proof is as follows.
|
|
Heisenberg's theory connects the solution of a problem in quantum mechanics with the, solution of a svstem of an infinite number of algebraic equations, in which the U:nknowns-infinite matrices-are allied to the classical position- and momentum-co-ordinates of the mechanical system, and functions of these, and obey peculiar calcul,ating rules. (The relation is this : to one position-, one momentumco-ordinate, or to one function of these corresponds always one infinite matrix.)
|
|
I will first show (§§ 2 and 3) how to each function of the positionand momentum-co-ordinates there may be related a matrix in such a manner, that these matrices, in every case, satisfy the formal calculating rules of Born and Heisenberg (among which I also reckon the so-called "quantum condition" or "interchange rule"; see below). This relation of matrices to functions is general; it takes no account of the special mechanical system considered, but is the same for all mechanical systems. (In other words : the particular Hamilton function does not enter into the connecting law.) However, the relation is still indefinite to a great extent. It arises, namely, from the auxiliary introduotion of an arbitrary complete orthogonal system of functions having for domain entire configuration space (N.B.-not
|
|
" pg-space ", but " q-space ' l The provisional indefiniteness of the
|
|
relation lies in the fact that we can assign the auxiliary role to an arbitrary orthogonal system.
|
|
After matrices are thus constructed in a very general way, so as to satisfy the general rules, I will show the following in § 4. The special system of algebraic equations, which, in a special case, connects the matrices of the position and impulse co-ordinates with the matrix of the Hamilton function, and which the authors call "equations of motion ", will be completely solved by assigning the auxiliary role to a de.finite orthogonal system, namely, to the system of proper functions of that partial differential equation which forms the basis of my wave mechanics. The solution of the natural boundary-value problem of this differential equation is completely equivalent to the solution of Heisenberg's algebraic problem. All Heisenberg's matrix elements, which
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|
1 My theory was inspired by L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925 (Theses, Paris, 1924), and by brief, yet infinitely far-seeing remarks of A. Einstein, Berl. Ber., 1925, p. 9 et seq. I did not at all suspect any relation to Heisenberg's theory at the beginning. I naturally knew about his theory, but was discouraged, if not repelled, by what appeared to me as very difficult methods of transcendental algebra, and by the want of perspicuity (Anschaulichkeit).
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MATRIX MECHANICS AND WAVE MECHANICS
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47
|
|
|
|
may interest us from the surmise that they define " transition probabilities " or "line intensities ", can be actually evaluated by differentiation and quadrature, as soon as the boundary-value problem is solved. Moreover, in wave mechanics, these matrix elements, or quantities that are closely related to them, have the perfectly clear significance of amplitude8 of the partial oscillations of the atom's electric moment. The intensity and polarisation of the emitted light is thus intelligible on the basis of the Maxwell-Lorentz theory. A. short preliminary sketch of this relationship is given in § 5.
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|
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|
§ 2. The Co-ordination of an Operator and of a Matrix with a Wellarranged Function-symbol and the Establishment of the Product Rule
|
|
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|
The starting-point in the construction of matrices is given by the
|
|
|
|
simple observation that Heisenberg's peculiar calculating laws for
|
|
|
|
functions of the double set of n quantities, qi, q2, ••• , q,,,, ; Pi, p 2, ..., Pn (position- and canonically conjugate momentum-co-ordinates)
|
|
|
|
agree exactly with the rules, which ordinary analysis makes linear
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|
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|
differential operators obey in the domain of the single set of n variables,
|
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|
|
aa· qi, q2, ••., qn. So the co-ordination has to occur in such a manner
|
|
that each pi in the function is to be replaced by the operator
|
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|
a · Actua11y
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t h e
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|
operator
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u~qz
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1s
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exc ha ngea bl e
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|
w·ith
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|
~O-,
|
|
uqm
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|
where m ~1zs
|
|
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arbitrary, but with qm only, if m-:t=l. The operator, obtained by
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interchange and subtraction when m = l, viz.
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(1)
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|
o-aqqz z -qzoaq-z,
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|
when applied to any arbitrary function of the q's, reproduces the function, i.e. this operator gives identity. This simple fact will be reflected in the domain of matrices as Heisenberg's interchange rule.
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|
After this· preliminary survey, we turn to systematic construction. Since, as noticed above, the interchangeability does not always hold good, then a definite operator does not correspond unique1y to a definite " function in the usual sense " of the q's and p's, but to a "function-symbol written in a definite way". Moreover, since we can perform only the operations of addition and multiplication with
|
|
the operators ~ , the function of the q's and p's must be written as a
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regular power s!ries in p at least, before we substitute u}qz for pz. It is sufficient to carry out the process for a single term of such a power series, and thus for a function of the following construction :
|
|
(2) F(qk, Pk}=f(qi • • • q,n)p,p,ptg(ql • • • qn}Pr' h(qi • • • qn)Pr#P,#· • •
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|
We wish to express this as a "well-arranged i function-symbol" and relate it to the following operator,
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1 Or " well-ordered."
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48
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WAVE MECHANICS
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|
..,
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|
wherein, somewhat more generally than in the preliminary survey,
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|
x!, Pr is not replaced by } simply, but by
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|
and K stands for a
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|
|
vqr
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|
Vlf.r
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|
universal constant. As an abbreviation for the operator arising out of
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the well-arranged function F, I have introduced the symbol [F, • ]
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in passing (i.e. only for the purpose of the present proof). The function
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(in the usual sense) of q1 .•. qn, which is obtained by using the operator on another function (in the usual sense), u(q1 .•• qn), will he denoted by [F, u]. If G is another well-arranged function, then
|
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[G F, u] will denote the function u after the operator of F has
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first been used on it, and then the operator of G; or, what is defined
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to be the same, when the operator of GF has been used. Of course
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this is not generally the same as [FG, u].
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Now we connect a matrix with a well-arranged function, like F,
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by means of its operator (3) and of an arbitrary complete orthogonal
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system having for its domain the whole of q-space. It is done
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as follows. For brevity we will simply write x for the group of
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f variables qi, q2, ••• qn, as is usual in the theory of Integral Equations,
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and write dx for an integral extending over the whole of q-space.
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The functions
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(4)
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|
u1(x)Vp(x}, u2(x)Vp(x}, u3(x)Vp(x} . . . ad inf.
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are now to form a complete orthogonal system, normalised to 1.
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{f Let, therefore, in every case
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{5)
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p(x)ui(x)uk(x)dx=O for i-4:k
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=l for i=k.
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Further, it is postulated that these functions vanish at the natural bound,ary of q-space (in general, infinity) in a way sufficient to cause the vanishing of certain boundary integrals which come in later on as secondary products after certain integrations by parts.
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|
By the operator (3) we now relate the following matrix,
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|
(6)
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Fkl=jp(x)uk(x)[F, ui(x)]dx,
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|
to the function F represented by (2). (The way of writing the indices on the left-hand side must not suggest the idea of "contravariance"; from this point of view, here discarded, one index was formerly written above, and the other below ; we write the matrix indices above, because later we will also have to write matrix elements, corresponding to the q's and p's, where the lower place is already occupied.) In words: a matrix element is computed by muUiplying the function of
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|
MAT.lU.X l\.l.1£<.J.tlA..l'HC~ A.NlJ WAV ~ lVrnlJtlAl''HlJi:;
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|
4~
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|
the orthogonal system denoted by the row-index (whereby we under-
|
|
stand always Ui, not u,rv'p), by the "density function" p, and by the result arising from using our operator on the orthogonal function corresponding to the column-index, and then by integrating the whole over the domain.1
|
|
It is not very difficult to show that additive and multiplicative combination of well-arranged functions or of the appertaining operators works out as matrix addition and matrix multiplication of the allied matrices. For addition the proof is trivial. For multiplication the proof runs as follows. Let G be any other well-arranged function, like F, and
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|
(7)
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|
the matrix corresponding. We wish to form the product matrix
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|
(FG)km = ~Fkl(]lm,
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|
l
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Before writing it, let us transform the expression (6) for Fkl as follows.
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|
By a series of integrations by parts, the operator [F, • ] is "revolved"
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|
from the function ui(x) to the function p(x)uk(x). By the expression
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|
"revolve " (instead of, say, " push ") I wish to convey that the
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|
sequence of the operations reverses itself exactly thereby. The
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|
|
boundary integrals, which come in as "by-products", are to disappear
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|
(see above). The "revolved " operator, including the change of
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|
sign that accompanies an odd number of differentiations, will be
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|
denoted by [F, • ]. For example, from (3) comes
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|
a2
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|
a
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|
(3') [F, • ] = ( - l)T . . . K 2u~qsh"uq(,,,q, 1 . . qn)K-u;qs,-,,
|
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|
a a a aa
|
|
g(ql . . . qn)K3 qt q, q,, f(q1 . . . qn),
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|
|
where ,- =number of differentiations. By applying this symbol, we have
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|
(6')
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|
If we now calculate the product matrix, we get {8} ~Fkl(]lm
|
|
l
|
|
7= {Jul(x)[F, p(x)uk(x)]dx. jp(x)uz(x)[G, um(x)]dx}
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|
= j[F, p(x)uk(x)][G, um(x)]dx.
|
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|
The last equation is simply the so-called "relation of completeness "
|
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|
1 More briefly: Fk1 is the kth "development coefficient" of the operator used on the function u 1•
|
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|
|
WA V.hi MEUHA.N !CS
|
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|
|
of our orthogonal system,1 applied to the " development coefficients ,, of the functions
|
|
1 [G, Um(x)] and p(xlF, p(x)uk(x)].
|
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|
|
Now in (8), let us revolve, by further integrations by parts, the
|
|
operator [.F, • ] from the function p(x)ui(x) back again to the function [G, Um(x)], so that the operator regains its original form. We clearly get
|
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|
|
(9)
|
|
|
|
(FG)km="fFkl()l,m= jp(x)uk(x)[FG, Um(x)]dx.
|
|
|
|
On the left is the (km)th element of the product matrix, and on the right, by the law of connection (6), stands the (km)th element of the matrix, corresponding to the well-arranged product FG. Q.E.D.
|
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|
|
§ 3. Heisenberg's Quantum Condition and the Rules for Partial Diflerentiation
|
|
|
|
Since operation {l) gave identity, then corresponding to the wellarranged function
|
|
|
|
( 10)
|
|
|
|
pzqz - qt[>z
|
|
|
|
we have the operator, multiplication by K, in accordance with our law of connection, in which we incorporated a universal constant K.
|
|
|
|
Hence to function (10) corresponds the matrix
|
|
|
|
(11)
|
|
|
|
f (pzqz-qt[>z)ik =K p(x)ui(x)uk(x)dx =0 for i -=t=k
|
|
|
|
=K for i=k.
|
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|
|
That is Heisenberg's "quantum relation" if we put
|
|
|
|
(12)
|
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|
|
K= h ,
|
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|
|
21rv-I
|
|
|
|
and this may be assumed to hold from now on. It is understood that
|
|
|
|
we could have also found relation (11) by taking the two matrices
|
|
|
|
f allied to qi and pz, viz.
|
|
|
|
(13)
|
|
|
|
qik = qzp(x)ui(x)uk(x)dx,
|
|
|
|
. J pi"k =K p(x)ui(x}O-Uakq(X;-)dx,
|
|
|
|
multiplying them together in different sequence and subtracting the two results.
|
|
Let us now turn to the "rules for partial differentiation". A well-arranged function, like (2), is said to be differentiated partially with respect to qz, when it is differentiated with respect to qz without
|
|
1 See, e.g., Courant-Hilbert, Metlwds of Mathematical Physics, I., p. 36. It is important to remember that the "relation of completeness " for the "development coefficients " is valid in every case, even when the developments themselves do not converge. If these do converge, then the equivalence (8) is directly evident.
|
|
|
|
MATRIX MECHANlCS A.NV WA V~ lVl.1!.;lJ.tlA.NlU~
|
|
|
|
01
|
|
|
|
altering the succession of the factors at each place where qz appears in it, and all these results are added.1 Then it is easy to show that the following equation between the operators is valid:
|
|
|
|
(14)
|
|
|
|
[ooqFz' • ] =jll-ptF-Fpi, • ].
|
|
|
|
The line of thought is this. Instead of really differentiating with respect to qi, it is very convenient simply to prefix Pl to the function;
|
|
as it is, pi must finally be replaced by K-::u:.0ql. Obviously I have to divide by K. Furthermore, when we apply the entire operator to any
|
|
|
|
function u, the operator 'u:lqoz will act not only on that part of F which contains ql (as it ought), but also wrongly on the function u, affected by the entire operator. This mistake is exactly corrected by subtracting again the operation [Fpz, • ] !
|
|
Consider now partial differentiation with respect to a Pl· Its
|
|
meaning for a well-arranged function, like (2), is a little simpler than
|
|
|
|
in the case of u}qz, because the p's only appear as power products. We imagine every power of Pl to be resolved into single factors, e.g. think of piptpz instead of pz3, and we can then say: in partial differentiation with respect to Pl, every separate pz that appears in Fis to be dropped once, all the other pz's remaining; all the results obtained are to be added. What will be the effect on the operator (3) 1
|
|
|
|
"Every separate K-::u:.q°z is to be dropped once, and all the results so obtained are to be added."
|
|
I maintain that on this reasoning the operational equation
|
|
|
|
(15)
|
|
|
|
[°oFpz, • _] =lK_[Fql -qzF, • ]
|
|
|
|
is valid. Actually, I picture the operator [Fql, • ] as formed and now attempt to " push qz through F from right to left ", that means, attempt to arrive at the operator [qtF, •] through successive exchanges. This pushing through meets an obstacle only as often as I come
|
|
|
|
against a aoqz. With the latter I may not interchange qi simply, but
|
|
|
|
have to replace
|
|
|
|
a
|
|
|
|
a
|
|
|
|
(16)
|
|
|
|
oqz by l +qzaqz
|
|
|
|
in the interior of the operator. The secondary products of the interchange, which are yielded by this " uniformising ", form just the
|
|
|
|
1 We a.re naturally following Heisenberg faithfully in all these definitionB. From a strictly logical standpoint the following proof is evidently superfluous, and we could have written down rules (14) and (15) right a.way, as they are proved in Heisenberg, and only depend upon the sum and product rules and the exchange rule (11) which
|
|
we have proved.
|
|
|
|
52
|
|
|
|
WAVE MECHANICS
|
|
|
|
desired " partial differential coefficients ", as is easily seen. After the pushing-through process is finished, the operator [qzF, • ] still remains left over. It would be superfluous and therefore is explicitly subtracted in (15). Hence (15) is proved. The equations (14) and (15), which have been proved for operators, naturally hold good unchanged for the matrices belonging to the right-hand and left-hand sides, because by (6) one matrix, and one on]y, belongs to one linear operator (after the system ui(x) has been chosen once for all).1
|
|
|
|
§ 4. The Solution of Heisenberg's Equations of Motion
|
|
We have now shown that matrices, constructed according to definitions (3) and (6) from well-arranged functions by the agency of an arbitrary, complete orthogonal system (4), satisfy all Heisenberg's calculating rules, including the interchange rule (11). Now let us consider a special mechanical problem, characterised by a definite Hamilton function
|
|
(17)
|
|
The authors of quantum mechanics take this function over from ordinary mechanics, which naturally does not give it in a "wellarranged" form; for in ordinary analysis no stress is laid on the sequence of the factors. They therefore " normalise " or " symmetricalise " the function in a definite manner for their purposes. For example, the usual mechanical function qkpk2 is replaced by
|
|
½(piq1c +q1cp-1c2)
|
|
|
|
1 In passing it may be noted that the converse of this theorem is also true, at least in the sense that certainly not more than one linear differential operator can belong to a given matrix, according to our connecting law (6), when the orthogonal system and
|
|
the density function are prescribed. For in (6), let the F 1d's be given, let [F, • J be the
|
|
linear operator we are seeking and which we presume to exist, and let ¢(x) be a function of q1, q2,, ••• , qn, which is sectionally continuous and differentiable as often as necessary, but otherwise arbitrary. Then the relation of completeness applied to the functions ¢(x) and [F, uk(x)] yields the following :
|
|
|
|
~{J jp(x)qi(x)[F, uk(x)]dx =
|
|
|
|
p(x)¢(x)u1(x)dx .jp(x)u,(x)[F, uk(x)]dx}·
|
|
|
|
The right-hand side can be regarded as definitely known, for in it occur only development coefficients of ¢(x) and the prescribed matrix elements F 1k. By "revolving" (see above), we can change the left-hand side into the kth development coefficient of the function
|
|
LF. p(x)¢(x)J
|
|
p(x) •
|
|
Thus all the development coefficients of this function are uniquely fixed, and thus so is the function itself (Courant-Hilbert, p. 37). Since, however, p(x) was fixed beforehand and ¢(x) is a quite arbitrary function, we can say: the result of the action of the revolved opera.tor on an arbitrary function, provided, of course, it can be submitted to the operator at all, is fixed uniquely by the matrix Flcl. This can only mean that the revolved operator is uniquely fixed, for the notion of" operator" is logically identical with the whole of the results of its action. By revolving the revolved operator, we ·obtain uniquely the operator we have sought, itself.
|
|
It is. to be noted that the developability of the functions which appear is not necessarily postulated-we have not proved that a linear opera.tor, corresponding to an arbitrary matrix, always exists.
|
|
|
|
MATRIX MECHANICS AND WAVE MECHANICS
|
|
|
|
53
|
|
|
|
or by
|
|
|
|
P"/1/kPk
|
|
|
|
or by
|
|
|
|
½(piqk + P"/1/kPk + qkpi),
|
|
|
|
which are all the same, according to (11). This function is then" well-
|
|
|
|
arranged ", i.e. the sequence of the factors is inviolable. I will not
|
|
|
|
enter into the general rule for symmetricalising here ; 1 the idea, if I
|
|
|
|
understand it aright, is that Hki is to be a diagonal matrix, and in
|
|
|
|
other respects the normalised function, regarded as one of ordinary
|
|
|
|
analysis, is to be identical with the one originally given. 2 We will
|
|
|
|
satisfy these demands in a direct manner.
|
|
|
|
Then the authors postulate that the matrices qzik, p/k shall satisfy
|
|
|
|
an infinite system of equations, as "equations of motion", and to
|
|
|
|
l _ begin with they write this system as follows :
|
|
|
|
( 18)
|
|
|
|
(d-qdtz)ik -_ (ou'.:IHpz)ik
|
|
|
|
l - l, 2, 3, . . . n
|
|
|
|
(ddptl)ik = ( - coHql)ik
|
|
|
|
.
|
|
i,
|
|
|
|
k
|
|
|
|
=
|
|
|
|
1,
|
|
|
|
2,
|
|
|
|
3,
|
|
|
|
ad. inf.
|
|
|
|
The upper pair of indices signifies, as before in Fki, the respective element of the matrix belonging to the well-arranged function in
|
|
i question. The meaning of the partial differential coefficient on the
|
|
right-hand side has just been explained, but not that of the appearing
|
|
|
|
on the left. By it the authors signify the foilowing. It is to give a
|
|
|
|
series of numbers
|
|
|
|
! (19)
|
|
|
|
v1, 112, v3, 114, . • . ad inf.,
|
|
|
|
such that the above equations are fulfilled, when to the is ascribed
|
|
|
|
the meaning: multiplication of the (ik)th matrix element by 27T~ (vi -11k)- Thus, in particular,
|
|
|
|
ddqtl)ik = 27Tv-T(vi - 11k)q/Tc;
|
|
|
|
(20)
|
|
|
|
l(,(ddptz)ik = 27Tv-T(J/i - llk)Plik.
|
|
|
|
The series of numbers (19) is not defined in any way beforehand, but
|
|
|
|
together with the matrix elements q/k, pz11c, they form the numerical
|
|
|
|
unlrnowns of the system of equations (18). The latter assumes the
|
|
|
|
form
|
|
|
|
fv; -v,)qi' -{( Hq, -q,H)
|
|
|
|
(18')
|
|
|
|
1(v; -v,)pi'-{( Hp,-p,H)
|
|
|
|
1 "Quantum Mechanics I." p. 873 et seq. 2 The stricter postulation-" shall yield the same quantum-mechanical equations of motion "-I consider too narrow. It arises, in my opinion, from the fact that the
|
|
authors confine themselves to power products with regard also to the qk's-which is unnecessary.
|
|
|
|
54
|
|
|
|
WAVE MECHANICS
|
|
|
|
when we utilise the explanation of the symbols (20), and the calculating rules (14) and (15), and take account of (12).
|
|
We must thus satisfy this system of equations, and we have no means at our disposal, other than the suitable choice of the orthogonal system (4), which intervenes in the formation of the matrices. I now assert the following :
|
|
1. The equations (18') will in general be satisfied if we choose as the orthogonal system the proper functions of the natural boundary value problem of the following partial differential equation,
|
|
|
|
(21)
|
|
|
|
-[H, rf,]+Erf,=0.
|
|
|
|
rf, is the unknown function of q1, q2, •.., qn; E is the proper value
|
|
parameter. Of course, as density function, p(x) appears that function of q1, . . ., qn, by which equation (21) must be multiplied in order to make it self-adjoint. The quantities vi are found to be equal to the proper values Ei divided by h. Hkl becomes a diagonal matrix, with H1ck=E1c.
|
|
2. If the symmetricalising of the function H has been effected in a suitable way-the process of symmetricalising, in my opinion, has not hitherto been defined uniquely-then (21) is identical with the wave equation which is the basis of my wave mechanics.1
|
|
Assertion 1 is almost directly evident, if we provisionally lay aside the questions whether equation (21) gives rise at all to an intelligible boundary value problem with the domain of entire q- space, and whether it can always be made self-adjoint through multiplication by a suitable function, etc. These questions are largely settled under heading 2. For now we have, according to (21) and the definitions of proper values and functions,
|
|
|
|
(22)
|
|
|
|
and thus from (6) we get
|
|
|
|
Hkl = jp(x)uk(x)[H, uz(x)]dx = Eijp(x)uk(x)uz(x)dx
|
|
|
|
(23)
|
|
|
|
1 =Oforl=t=k =Ezfor l=k,
|
|
|
|
and, for example,
|
|
|
|
( (Hqz)ik = ~Himqrk = Eiqik
|
|
|
|
(24)
|
|
|
|
l(qzH)ik = ~qzimHmlc·= E,..qilc,
|
|
|
|
m
|
|
|
|
so that the right-hand side of the first equation of (18') takes the value
|
|
|
|
(25)
|
|
|
|
Similarly for the second equation. Thus everything asserted under I is proved.
|
|
1 Equation (18"), Part II.
|
|
|
|
MATRIX MECHANICS AND WAVE MECHANICS
|
|
|
|
55
|
|
|
|
Let us tum now to assertion 2, which is, that there is agreement between the negatively taken operator of the Hamilton function (suitably symmetricalised) and the wave operator of wave mechanics. I wi11 first illustrate by a simple example why the process of symmetricalisation seems to me to be, in the first instance, not unique. Let, for one degree of fr~edom, the ordinary Hamilton function be
|
|
|
|
(26)
|
|
|
|
H = ½(p2 + q2).
|
|
|
|
Then it is admitted that we can take this function, just as it stands, unchanged, over to "quantum mechanics" as a "well-arranged" function. But we can also, and seemingly indeed with as much right to begin wil,k, apply the well-arranged function
|
|
|
|
(27)
|
|
|
|
H = ½~~l(q)p +q"),
|
|
|
|
where/(q) is a function arbitrary within wide limits. f(q) would appear in this case as a "density function " p(x). (26) is quite evidently just a special case of (27), and the question arises, whether (and how) it is at all possible to distinguish the special case we are concerned with, i.e. for more complicated H-functions. Confining ourselves to pow~r products only of the qk's (where we could then simply prohibit the " Rroduction of denominators ") would be most inconvenient just in the ,most important applications. Besides, I believe that does not lead to correct symmetricalisation.
|
|
For the convenience of the reader, I will now give again a short derivation of the wave equation in a form suited to the present purpose, confining myself to the case of classical mechanics (without relativity and magnetic fields). Let, therefore,
|
|
|
|
(28)
|
|
|
|
H = T(qk, Pk)+ V(qk),
|
|
|
|
T being a quadratic form in the pk's. Then the wave equation can be deduced 1 from the following variation problem,
|
|
|
|
&11 =8!{t>(qb : ) +tp2V(q.) }Lip-idx=O,
|
|
|
|
(29)
|
|
|
|
with the subsidiary condition
|
|
|
|
{
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J2 = ft,2~ -idx = I.
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J f ... f As above, dx stands for
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dq1 •.• dqn; dp-l is the reciprocal
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of the square root of the discriminant of the quadratic form T. This factor must not be omitted, because otherwise the whole process would not be invariant for point transformations of the q's ! By all means
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another explicit function of the q's might appear as a factor, i.e. a function which would be invariant for a point transformation of the q's. (For dp, as is known, this is not the case. Otherwise we coul,d omit dp -•, if this extra function was given the value di.)
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If we indicate the derivative of T with respect to that argument,
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1 Equations (23) and (24) of Part I.
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56
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WAVE MECHANICS
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which originally was Pk, by the suffix Pk, we obtain, as the result of the
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variation,
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O=½(8J1 - E8J2)
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(30)
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~J =J{ -::, f :t.)J dp-•Tp,(q~
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+ (V(q•) - E)dp-trp}a,f,dx;
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the Eulerian variation equation thus runs :
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(31)
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f oifi)} a{ ( sh1r22L\i aqk L\p - •TP1; qk, oqk - V(qk)t/J + Ey, = o.
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It is not difficult to see that this equation has the form of (21) if we remember our law connecting the operators, and consider
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(32)
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T(qk, p1c) = ½Lp1cTP (% p1c)
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le
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J;
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the Eulerian equation for homogeneous functions, applied to the
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quadratic form T. In actual fact, if we detach the operator from
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the left side of (31), with the proper value term Ey, removed, and
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replace
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in
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it
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~
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27T -1
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u~qa k
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by P>,
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then
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according
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to
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(32)
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we
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obtain
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the
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negatively taken Hamilton function (28). Thus the process of variation
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has given quite automatically a uniquely defined " symmetricalisation "
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of the operator, which makes it self-adjoint (except possibly for a
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common factor) and makes it invariant for point transformations, and
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which I would like to maintain, as long as there are no definite reasons
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for the appear2.nce under the integrals (29) of the additional factor,
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already 1 mentioned as possible, and for a definite form of the latter.
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Hence the solution of the whole system of matrix equations of
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Heisenberg, Born, and Jordan is reduced to the natural boundary
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value problem of a linear partial differential equation. If we have
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solved the boundary value problem, then by the use of (6) we can
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calculate by differentiations and quadratures every matrix element we
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are interested in.
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As an illustration of what is to be understood by the natural
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boundary value problem, i.e. by the natural boundary conditions at
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the natural boundary of configuration space, we may refer to the
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worked examples.2 It invariably turns out that the natural infinitely
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distant boundary forms a singularity of the differential equation and
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only allows of the one boundary condition-" remaining finite ". This
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seems to be a general characteristic of those micro-mechanical prob-
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lems with which the theory in the first place is meant to deal. If the
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domain of the position co-ordinates is artificially limited (example :
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a molecule in a "vessel"), then an essential allowance must be made
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for this limitation by the introduction of suitable potential energies in
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1 Cf. also Ann. d. Phys. 79, p. 362 and p. 510 (i.e. Parts I. and 11.). 2 In Parts I. and II. of this collection.
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MATRIX MECHANICS AND WAVE MECHANICS
|
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57
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|
the well-known manner. Also the vanishing of the proper functions at the boundary generally occurs to an adequate degree, even if relations among certain of the integrals (6) are present, which necessitate a special investigation, and into which I will not enter at present. (It. has to do with those matrix elements in the Kepler problem which, according to Heisenberg, correspond to the transition from one hyperbolic orbit to another.)
|
|
I have confined myself here to the case of classical mechanics, without magnetic fields, because the relativistic magnetic generalisa-tion does not seem to me to be sufficiently clear yet. But we can scarcely doubt that the complete parallel between the two new quantum theories will still stand when this generalisation is obtained.
|
|
We conclude with a general observation on the whole formal apparatus of§§ 2, 3, and 4. The basic orthogonal system was regarded as an absolutely discrete system of functions. Now, in the most important applications this is not the case. Not only in the hydrogen atom but also in heavier atoms the wave equation (31) must possess a continuous proper value spectrum as well as a line spectrum. The former manifests itself, for example, in the continuous optical spectra which adjoin the limit of the series. It appeared better, provisionally, not to burden the formulae and the line of thought with this generalisation, though it is indeed indispensable. The chief aim of this paper is to work out, in the clearest manner possible, the formal connection between the two theories, and this is certainly not changed, in any essential point, by the appearance of a continuous spectrum. An important precaution that we have always observed is not to postulate, without further investigation, the convergence oi the development in a series of proper functions. This precaution is especially demanded by the accumul,ation of the proper values at a finite point (viz. the limit of the series). This accumulation is most intimately connected with the appearance of the continuous spectrum.
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§ 5. Comparison of the Two Theories. Prospect of a Classical Understanding of the Intensity and Polarisation of the Emitted Radiation
|
|
If the two theories-I might reasonably have used the singularshould 1 be tenable in the form just given, i.e. for more complicated systems as well, then every discussion of the superiority of the one over the other has only an illusory object, in a certain sense. For they are completely equivalent from the mathematical point of view, and it can only be a question of the subordinate point of convenience of calculation.
|
|
1 There is a. special reason for leaving this question open. The two theories initially take the energy function over from ordinary mechanics. Now in the cases treated the potential energy arises from the interaction of particles, of which perhaps one, at least, may be regarded in wave mechanics also as forming a point, on account of its great mass (cf. A. Einstein, Berl. Ber., 1925, p. 10). We must take into account the possibility that it is no longer permissible to take over from ordinary mechanics the statement for the potential energy, if both " point charges "a.re really extended states of vibration, which penetrate ea.ch other.
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58
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WAVE MECHANICS
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|
To-day there are not a few physicists who, like Kirchhoff and Mach, regard the task of physical theory as being merely a mathematical description (as economical as possibl,e) of the empirical connections between observable quantities, i.e. a description which reproduces the connection, as far as possible, without the intervention of unobservable elements. On this view, mathematical equivalence has almost the same meaning as physical equivalence. In the present case there might perhaps appear to be a certain superiority in the matrix representation because, through its stifling of intuition, it does not tempt us to form space-time pictures of atomic processes, which must perhaps remain uncontrollable. In this connection, however, the following supplement to the proof of equivalence _given above is interesting. The equivalence actually exists, and it also exists conversely. Not only can the matrices be constructed from the proper functions as shown above, but also, conversely, the functions can be constructed from the numerically given matrices. Thus the functions do not form, as it were, an arbitrary and special " fleshly dothing,, for the bare matrix skeleton, provided to pander to the need for intuitiveness. This really would establish the superiority of the matrices, from the epistemological point of view. We suppose that in the equations
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|
(33)
|
|
|
|
the left-hand sides are given numerically and the functions Ui( x) are to be found. ( N.B.-The "density function" is omitted for simplicity; the ui(x)'s themselves are to be orthogonal functions for the present.) We may then calculate by matrix multiplication (without, by the way, any "revolving", i.e. integration by parts) the following integrals,
|
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|
|
(34)
|
|
|
|
where P(x) signifies any power product of the q/s. The totality of these integrals, when i and k are fixed, forms what is called the totality of the "moments" of the function ui(x)uk(x). And it is known that, under very general assumptions, a function is determined uniquely by the totality of its moments. So all the products ui(x)u1c(x) are uniquely fixed, and thus also the squares u.i(x)2, and therefore also ui( x) itself. The only arbitrariness lies in the supplementary detachment of the density function p(x), e.g. r2 sin 0 in polar co-ordinates. No false step is to be feared there, certainly not so far as epistemology is concerned.
|
|
Moreover, the validity of the thesis that mathematical and physical equivalence mean the same thing, must itself be qualified. Let
|
|
J us think, for example, of the two expressions for the electrostatic
|
|
energy of a system of charged conductors, the space integral ½ E2d-r
|
|
and the sum ½~CiVi taken over the conductors. The two expressions
|
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|
|
MATRIX MECHANICS AND WAVE MECHANICS
|
|
|
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59
|
|
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|
are completely equivalent in electrostatics; the one may be derived from the other by integration by parts. Nevertheless we intentionally prefer the first and say that it correctly localises the energy in space. In the domain of electrostatics this preference has admittedly no justification. On the contrary, it is due simply to the fact that the first expression remains useful in electrodynamics also, while the second does not.
|
|
We cannot yet say with certainty to which of the two new quantum theories preference should be given, from this point of view. As the natural advocate of one of them, I will not be blamed if I franklyand perhaps not wholly impartially-bring forward the arguments in its favour.
|
|
Leaving aside the special optical questions, the problems which the course of development of atomic dynamics brings up for consideration are presented to us by experimental physics in an eminently intuitive form ; as, for example, how two colliding atoms or molecules rebound from one another, or how an electron or a-particle is diverted, when it is shot through an atom with a given velocity and with the initial path at a given perpendicular distance from the nucleus. In order to treat such problems more particularly, it is necessary to survey clearly the transition between macroscopic, perceptual mechanics and the micro-mechanics of the atom. I have lately 1 explained how I picture this transition. Micro-mechanics appears as a refinement of macro-mechanics, which is necessitated by the geometrical and mechanical smallness of the objects, and the transition is of the same nature as that from geometrical to physical optics. The latter is demanded as soon as the wave length is no longer very great compared with the dimensions of the objects investigated or with the dimensions of the space inside which we wish to obtain more accurate information about the light distribution. To me it seems extraordinarily difficult to tackle problems of the above kind, as long as we feel obliged on epistemological grounds to repress intuition in atomic dynamics, and to operate only with such abstract ideas as transition probabilities, energy levels, etc.
|
|
An especially important question-perhaps the cardinal question of all atomic dynamics-is, as we know, that of the coupling between the dynamic process in the atom and the electromagnetic field, or whatever has to appear in the place of the latter. Not only is there connected with this the whole complex of questions of dispersion, of resonanceand secondary-radiation, and of the natural breadth of lines, but, in addition, the specification of certain quantities in atomic dynamics, such as emission frequencies, line intensities, etc., has only a mere dogmatic meaning until this coupling is described mathematically in some form or other. Here, now, the matrix representation of atomic dynamics has led to the conjecture that in fact the electromagnetic field also must be represented otherwise, namely, by matrices, so that the coupling may be mathematically formulated. Wave mechanics
|
|
1 Part II.
|
|
|
|
60
|
|
|
|
WAVE MECHANICS
|
|
|
|
shows we are not compelled to do this in any case, for the mechanical field scalar (which I denote by y,) is perfectly capable of entering into the unchanged Maxwell-Lorentz equations between the electromagnetic field vectors, as the "source" of the latter; just as, conversely, the electrodynamic potentials enter into the coefficients of the wave equation, which defines the field scalar. 1 In any case, it is worth while attempting the representation of the coupling in such a way that we bring into the unchanged Maxwell-Lorentz equations as four-current a four-dimensional vector, which has been suitably derived from the mechanical field scalar of the electronic motion (perhaps through the medium of the field vectors themselves, or the potentials). There even exists a hope that we can represent the wave equation for if, equally well as a consequence of the Maxwell-Lorentz equations, namely, as an equation of continµity for electricity. The difficulty in regard to the problem of several electrons, which mainly
|
|
lies in the fact that tf, is a function in configuration space, not in real
|
|
space, must be mentioned. Nevertheless I would like to discuss the one-electron problem a little further, showing that it may be possible to give an extraordinarily clear interpretation of intensity and polarisation of radiation in this manner.
|
|
Let us consider the picture, on the wave theory, of the hydrogen
|
|
atom, when it is in such a state that the field scalar tf, is given by a
|
|
series of discrete proper functions, thus :
|
|
|
|
z,,.~Et
|
|
|
|
(35)
|
|
|
|
if,=~ckuk(x)e-h- k
|
|
|
|
k
|
|
|
|
(x stands here for three variables, e.g. r, 0, <p; the ck's are taken as real
|
|
|
|
and it is correct to take the real part). We now make the assumption
|
|
|
|
that the space density of electricity is given by the real part of
|
|
|
|
(36)
|
|
|
|
if at
|
|
|
|
The bar is to denote the conjugate complex function. We then calculate for the space density,
|
|
|
|
(37) space density= 21r ~ ckcmEk~ Emuk(x)it,,n(x) sin 2h77t(Em -Ek),
|
|
(k,m)
|
|
where the sum is to be taken once only over every combination (k, m). Only term differences enter (37) as frequencies. The former are so low that the length of the corresponding ether wave is large compared
|
|
|
|
1 Similar ideas are expressed by K. Lanczos in an interesting note that has just
|
|
appeared (Ztschr. f. Phys. 35, p. 812, 1926). This note is also valuable as showing
|
|
that Heisenberg's atomic dynamics is capable of a continuous interpretation as well. However, Lanczos' work has fewer points of contact with the present work than at first it was thought to have. The determination of his formal system, which was provisionally left quite indefinite, is not to be sought by following the idea that in some way the symmetrical nucleus K (s, tr) of Lanczos can be identified with the Green's junction of our wave equation (21) or (31). For this Green's function, if it exists, has the quantum levels themselves as proper values. On the other hand, it is required that Lanczos' function should have the reciprocals of the quantum levels as proper
|
|
values.
|
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|
|
lVLA'l'Hl.X. M.h:UHA.NlCS AND WAVE MECHANICS
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61
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|
|
|
with atomic dimensions, that is, compared with the region within which (37) is markedly different from zero. 1 The radiation can there-
|
|
|
|
fore be estimated simply by the dipole moment which according to (37) the whole atom possesses. We multiply (37) by a Cartesian co-ordinate ql, and by the "density function" p(x), (r2 sin 0 in the present case) and integrate over the whole space. According to (13), we get for the component of the dipole moment in the direction ql,
|
|
|
|
(3 8 )
|
|
|
|
Mqi= 21r ~~ ckcmqJrr.mEk-hEm sm. 2-ThTt(Em- Ek).
|
|
|
|
(k,m)
|
|
|
|
Thus we really get a "Fourier development" of the atom's electric moment, in which only term differences appear as frequencies. The Heisenberg matrix elements qfm come into the coefficients in such a manner that their co-operating influence on the intensity and polarisation of the part, of the radiation concerned is completely intelligible on the grounds of classical electrodynamics.
|
|
The present sketch of the mechanism of radiation is far from completely satisfactory and is in no way final. Assumption (36) makes use, somewhat freely, of complex calculation, in order to put to one side undesired components of vibration whose radiation cannot be investigated at all in the simple way used for the dipole moment of the entire atom, because the corresponding ether wave lengths (about
|
|
|
|
0·01 A) lie far below atomic dimensions. Moreover, if we integrate over all space, then by (5) the space density (37) gives zero and not, as is required, a finite value, independent of the time, which requires to be normalised to the electronic charge. In conclusion, for completeness, account should be taken of magnetic radiation, since if there is a spatial distribution of electric currents, radiation is possible without the appearance of an electric moment, e.g. with a frame aerial.
|
|
Nevertheless it appears to be a well-founded hope that a real understanding of the nature of emitted radiation will be obtained on the basis of one of the two very similar analytical mechanisms which have been sketched here.
|
|
|
|
(Received March 18, 1926).
|
|
1 Ann. d. Phys. 79, p. 371, 1926, i.e. beginning of§ 2, Part I. here.
|
|
|
|
Quantisation as a Problem of
|
|
Proper Values (Part III)
|
|
PERTURBATION THEORY, WITH APPLICATION TO THE STARK EFFECT
|
|
OF THE BALMER LINES
|
|
(Annalen der Physik (4), vol. 80, 1926)
|
|
Introduction. Abstract
|
|
As has already been mentioned at the end of the preceding paper,1 the available range of application of the proper value theory can by comparatively elementary methods be considerably increased beyond the " directly soluble problems " ; for proper values and functions can readily be approximately determined for such boundary value problems as are sufficiently closely related to a directly soluble problem. In analogy with ordinary mechanics, let us call the method in question the perturbation method. It is based upon the important property of continuity possessed by proper values and functions, 2 principally, for our purpose, upon their continuous dependence on the coefficients of the differential equation, and less upon the extent of the domain and on the boundary conditions, since in our case the domain (" entire q-space ") and the boundary conditions (" remaining finite ") are generally the same for the unperturbed and perturbed problems.
|
|
The method is essentially the same as that used by Lord Rayleigh in investigating 3 the vibrations of a string with small inhomogeneities in his Theory of Sound (2nd edit., vol. i., pp. 115-118, London, 1894). This was a particularly simple case, as the differential equation of the unperturbed problem had constant coefficients, and only the perturbing terms were arbitrary functions along the string. A complete generalisation is possible not merely with regard to these points, but also for the specially important case of several independent variables, i.e. for partial differential equations, in which multiple proper values appear in the. unperturbed problem, and where the addition of a
|
|
1 Last two paragraphs of Part II. 2 Courant-Hilbert, chap. vi. §§ 2, 4, p. 337. 3 Courant-Hilbert, chap. v. § 5, 2, p. 241.
|
|
62
|
|
|
|
\lUAl~Tl~ATlUl~ A.NlJ YtW.1'1£1{ VALUES-III
|
|
|
|
63
|
|
|
|
perturbing term causes the splitting up of such values and is of the greatest interest in well- known spectroscopic questions (Zeeman effect, Stark effect, Multiplicities). In the development of the perturbation theory in the following Section I., which really yields nothing new to the mathematician, I put less value on generalising to the widest possible extent than on bringing forward the very simple rudiments in the clearest possible manner. From the latter, any desired generalisation arises almost automatically when needed. In Section II., as an example, the Stark effect is discussed and, indeed, by two methods, of which the first is analogous to Epstein's method, by which he first solved 1 the problem on the basis of classical mechanics, supplemented by quantum conditions, while the second, which is much more general, is analogous to the method of secular perturbations.2 The first method will be utilised to show that in wave mechanics also the perturbed problem can be "separated" in parabolic co-ordinates, and the perturbation theory will first be applied to the ordinary differential equations into which the original vibration equation is split up. The theory thus merely takes over the task which on the old theory devolved on Sommerfeld's elegant complex integration for the calculation of the quantum integrals. 3 In the second method, it is found that in the case of the Stark effect an exact separation coordinate system exists, quite by accident, for the perturbed problem also, and the perturbation theory is applied directly to the partial differential equation. This latter proceeding proves to be more troublesome in wave mechanics, although it is theoretically superior, being more capable of generalisation.
|
|
Also the problem of the intensity of the components in the Stark effect will be shortly discussed in Section II. Tables will be calculated, which, as a whole, agree even better with experiment than the wellknown ones calculated by Kramers with the help of the correspondence princi pie. 4
|
|
The application (not yet completed) to the Zeeman effect will naturally be of much greater interest. It seems to be indissolubly linked with a correct formulation in the language of wave mechanics of the relativistic problem, because in the four-dimensional formulation the vector-potential automatically ranks equally with the scalar. It was already mentioned in Part I. that the relativistic hydrogen atom may indeed be treated without further discussion, but that it leads to "half-integral " azimuthal quanta, and thus contradicts experience. Therefore " something must still be missing ". Since then I have learnt what is lacking from the most important publications of G. E. Uhlenbeck and S. Goudsmit,5 and then from oral and written communications from Paris (P. Langevin) and Copenhagen (W. Pauli),
|
|
|
|
1 P. S. Epstein, Ann. d. Phys. 50, p. 489, 1916. 2 N. Bohr, Kopen"hagener Akademie (8), IV., 1, 2, p. 69 et seq., 1918. 3 A. Sommerfeld, Atombau, 4th ed., p. 772. 4 H. A. Kramers, Kopenhagener Akademie (8), III., 3, p. 287, 1919. 6 G. E. Uhlenbeck and S. Goudsmit, Physica, 1925; Die Naturwissenschafte11,
|
|
1926; Nature, 20th Feb., 1926; cf. also L. H. Thomas, Nature, 10th April, 1926.
|
|
|
|
64
|
|
|
|
WAVE MECHANICS
|
|
|
|
viz., in the language of the theory of electronic orbits, the angul,ar momentum of the electron round its axis, which gives it a magnetic moment. The utterances of these investigators, together with two highly significant papers by Slater 1 and by Sommerfeld and Unsold 2 dealing with the Balmer spectrum, leave no doubt that, by the introduction of the paradoxical yet happy conception of the spinning electron, the orbital theory will be able to master the disquieting difficulties which have latterly begun to accumulate (anomalous Zeeman effect; PaschenBack effect of the Balmer lines ; irregular and regular Rontgen doublets; analogy of the latter with the alkali doublets, etc.). We shall be obliged to attempt to take over the idea of Uhlenbeck and Goudsmit into wave mechanics. I believe that the latter is a very fertile soil for this idea, since in it the electron is not considered as a point charge, but as continuously flowing through space,3 and so the unpleasing conception of a "rotating point-charge" is avoided. In the present paper, however, the taking over of the idea is not yet attempted.
|
|
To the third section, as "mathematical appendix", have been relegated numerous uninteresting calculations-mainly quadratures of products of proper functions, required in the second section. The formulae of the appendix are numbered (101), (102), etc.
|
|
|
|
I. PERTURBATION THEORY
|
|
|
|
§ I. A Single Independent Variable
|
|
|
|
Let us consider a linear, homogeneous, differential expression of the second order, which we may assume to be in self-adjoint form without loss of generality, viz.
|
|
|
|
(1)
|
|
|
|
L[y] =py" +p'y' -qy.
|
|
|
|
y is the dependent function ; p, p' and q are continuous functions of the independent variable x and p-:?.0. A dash denotes differentiation with respect t_o x (p' is therefore the derivative of p, which is the condition for self-adjointness).
|
|
Now let p(x) be another continuous function of x, which never becomes negative, and also in general does not vanish. We consider the proper value problem of Sturm and Liouville,4
|
|
|
|
(2)
|
|
|
|
L[y] + E py = 0.
|
|
|
|
It is a question, first, of finding all those values of the constant E (" proper values") for which the equation (2) possesses solutions y(x), which are continuous and not identically vanishing within a certain domain, and which satisfy certain "boundary conditions" at the bounding points ; and secondly of finding these solutions (" proper
|
|
|
|
1 J. C. Slater, Proc. Amer. Nat. Acad. II, p. 732, 1925.
|
|
2 A. Sommerfeld and A. Unsold, Ztachr. J. Phys. 36, p. 259, 1926.
|
|
3 Cf. Jast two pages of previous paper.
|
|
4 Cf. Courant-Hilbert, chap. v. § 5, I, p. 238 et seq.
|
|
|
|
QUANTISATION AND PROPER VALUES-III
|
|
|
|
65
|
|
|
|
functions") themselves. In the cases treated in atomic mechanics, domain and boundary conditions are always" natural". The domain, for example, reaches from 0 to oo, when x signifies the value of the radius vector or of an intrinsically positive parabolic co-ordinate, and the boundary conditions are in these cases : remaining finite. Or, when x signifies an azimuth, then the domain is the interval from 0 to 21r and the condition is : Repetition of the initial values of y and y' at the end of the interval (" periodicity ").
|
|
It is only in the case of the periodic condition that multiple, viz. double-valued, proper values appear for one independent variable. By this we understand that to the same proper value belong several (in the particular case, two) linearly independent proper functions. We will now exclude this case for the sake of simplicity, as it attaches itself easily to the developments of the following paragraph. Moreover, to lighten the formulae, we will not expressly take into account in the notation the possibility that a "band spectrum" (i.e. a continuum of proper values) may be present when the domain extends to infinity.
|
|
Let now y =Ui(x), i = l, 2, 3, ..., be the series of Sturm-Liouville
|
|
proper functions; then the series of functions ui(x)'\l'p(x[i = I, 2, 3, ..., forms a complete orthogonal system for the domain ; i.e. in the first place, if ui(x) and uk(x) are the proper functions belonging to the values Ei and Ek, then
|
|
|
|
(3)
|
|
|
|
jp(x)ui(x)uk(x)dx=O for i *k.
|
|
|
|
(Integrals without limits are to be taken over the domain, throughout this paper.) The expression "complete " signifies that an originally arbitrary continuous function is condemned to vanish identically, by the mere postulation that it must be orthogonal with respect to -all the
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|
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functions ui(x)Vp(x}. (More shortly: "There exists no further orthogonal function for the system.") We can and will always regard the
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proper functions ui(x) in all general discussions as "normalised", i.e. we imagine the constant factor, which is still arbitrary in each of them on account of the homogeneity of (2), to be defined in such a way that the integral (3) takes the value unity for i = k. Finally we again
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remind the reader that the proper values of (2) are certainly all real. Let now the proper values Ei and functions ui(x) be known. Let
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us, from now on, direct our attention specially to a definite proper
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value, Ek say, and the corresponding function uk(x), and ask how these alter, when we do not alter the problem in any way other than by adding to the left-hand side of (2) a small "perturbing term", which we will initially write in the form
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(4)
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-Ar(x)y.
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In this A is a small quantity (the perturbation parameter), and r( x) is an arbitrary continuous function of x. It is therefore simply a matter of a slight alteration of the coefficient q in the differential
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expression (1). From the continuity properties of the proper quantities,
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66
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WAVE MECHANICS
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mentioned in the introduction, we now know that the altered SturmLiouville problem
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(2')
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L[y]-Ary + Epy=O
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must have, in any case for a sufficiently small A, proper quantities in the near neighbourhood of Ek and uk, which we may write, by way of trial, as
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(5)
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Ek* =E1c +AE:.i;; uk* =uk(x) +Avk(x).
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On substituting in equation (2'), remembering that uk satisfies (2), neglecting A2 and cutting away a factor Awe get
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(6)
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L[vk]+Ekpvk=(r-E:kp)uk.
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For the defining of the perturbation vk of the proper function, we thus obtain, as a comparison of (2) and (6) shows, a non-horrwgeneous equation, which belongs precisely to that homogeneous equation which is satisfied by our unperturbed proper function Uk (for in (6) the special proper value Ek stands in place of E). On the right-hand side of this non-homogeneous equation occurs," in addition to known quantities, the still unknown perturbation €k of the proper value.
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This occurrence of €k serves for the calculation of this quantity before the calculation of vk. It is known that the non-homogeneous equation-and this is the starting-point of the whole perturbation theory -for a proper value of the homogeneous equation possesses a solution when, and only when, its right-hand side is orthogonal 1 to the allied proper function (to all the allied functions, in the case of multiple proper values). (The physical interpretation of this mathematical theorem, for the vibrations of a string, is that if the force is in resonance with a proper vibration it must be distributed in a very special way over the string, namely, so that it does no work in the vibration in question; otherwise the amplitude grows beyond all limits and a stationary condition is impossible.)
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The right-hand side of (6) must therefore be orthogonal to uk,
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(7)
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or
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Jruk2dx
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(7')
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€k=---,
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fpuk2dx
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or, if we imagine Ui already normalised, then, more simply,
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(7")
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€k = jruidx.
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This simple formula expresses the perturbation of the proper value (of first order) in terms of the perturbing function r(x) and the unperturbed proper function uk(x). If we consider that the proper
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1 Cf. Courant-Hilbert, chap. v. § IO, 2, p. 277.
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QUANTISATION AND PROPER VALUES-III
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67
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value of our problem signifies mechanical energy or is analogous to it, and that the proper function uk is comparable to " motion with energy E,. ", then we see in (7") the complete parallel to the wellknown theorem in the perturbation theory of classical mechanics, viz. the perturbation of the energy, to a first approximation, is equal to the perturbing function, averaged over the unperturbed motion. (It may be remarked in passing that it is as a rule sensible, or at least aesthetic, to throw into bold relief the factor p(x) in the
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;i:i integrands of all integrals taken over the entire domain. If we do this,
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then, in integral (7"), we must speak of and not r(x) as the perturb-
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ing function, and make a corresponding change in the expression (4). Since the point is quite unimportant, however, we will stick to the notation already chosen.)
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We have yet to define vk(x), the perturbation of the proper function, from (6). We solve 1 the non-homogeneous equation by putting for vk a series of proper functions, viz.
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CX)
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(8)
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vk(x) = ~ 'Ykiui(x),
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i=l
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and by developing the right-hand side, divided by p(x), likewise in a
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series of proper functions, thus
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(9) where
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r(x)
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(
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p -
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()
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X
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-
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) E1c uk(x) =
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cxi
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.~ ckiui(x),
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t=l
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cki = j(r - Ekp)ukutdx
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f ( = rukuidx for i =I= k
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=0
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for i =k.
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The last equality follows from (7). If we substitute from (8) and (9)
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in (6) we get
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'(11)
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00
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CX)
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~ 'Yki(L[ui] + E1tpui) = ~ CkiP'Ui-
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i= l
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i=l
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Since now Ui satisfies equation (2) with E = Ei, it follows that
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(12)
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00
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CX)
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~ 'YkiP(Ek - Ei)Ui = ~ CkiPUi,
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i=l
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i=l
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By equating coefficients on left and right, all the 'Yki's, except 'Ykk, are
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defined. Thus
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(13)
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frukuidx
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Yki =
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E Cki
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k-
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E
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i
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=
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-E-=---=E--
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k- i
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f or
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•
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i =I=
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k
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,
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while Ykk, as may be understood, remains completely undefined. This indefiniteness corresponds to the fact that the postulation of
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1 Cf. Courant-Hilbert, chap. v. § 5, 1, p. 240, and§ 10, p. 279.
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68
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|
WA VE MECHANICS
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normalisation is still available for us for the perturbed proper
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function. If we make use of (8) in (5) and claim for uk*(x) the same
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normalisation as for uk(x) (quantities of the order of ..\2 being
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neglected), then it is evident that Ykk=O. Using (13) we now obtain
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for the perturbed proper function
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f oo Ui( x) rukuidx
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(14)
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uk*(x) =uk(x) +,,\ i~: Ek_ Ei •
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(The dash on the sigma denotes that the term i =k has not to be taken.) And the allied perturbed proper value is, from the above,
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(15)
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|
Ek*= Ek+,,\jru2~x.
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By substituting in (2') we may convince ourselves that (14) and (15) do really satisfy the proper value problem to the proposed degree of approximation. This verification is necessary since the development, assumed in (5), in integral powers of the perturbation parameter is no necessary consequence of continuity.
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|
The procedure, here explained in fair detail for the simplest case, is capable of generalisation in many ways. In the first place, we can of course consider the perturbation in a quite similar manner for the second, and then the third order in..\, etc., in each case obtaining first the next approximation to the proper value, and then the corresponding approximation for the proper function. In certain circumstances it may be advisable-just as in the perturbation theory of mechanicsto regard the perturbation function itself as a power series in..\, whose terms come into play one by one in the separate stages. These questions are discussed exhaustively by Herr E. Fues in work which is now appearing in connection with the application to the theory of band spectrr.
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In the second place, in quite similar fashion, we can consider also a perturbation of the term in y' of the differential operator (I) just as we have considered above the term - qy. The case is important, for the Zeeman effect leads without doubt to a perturbation of this kind-though admittedly in an equation with several independent variables. Thus the equation loses its self-adjoint form by the perturbation-not an essential matter in the case of a single variable. In a partial differential equation, however, this loss may result in the perturbed proper values no longer being real, though the perturbing term is real; and naturally also conversely, an imaginary perturbing term may have a real, physically intelligible perturbation as its consequence.
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|
We may also go further and consider a perturbation of the term in y". Indeed it is quite possible, in general, to add an arbitrary " infinitely smaJl " linear 1 and homogeneous differential operator, even of higher order than the second, as the perturbing term and to calculate the perturbations in the same manner as above. In these cases,
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1 Even the limitation "linear" is not absolutely necessary.
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|
QUANTISATION AND PROPER VALUES-III
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|
69
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|
however, we would use with advantage the fact that the second and higher derivatives of the proper functions may be expressed by means of the differential equation itself, in terms of the zero and first derivatives, so that this general case may be reduced, in a certain sense, to the two special cases, first considered-perturbation of the terms in y and y'.
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|
Finally, it is obvious that the extension to equations of order higher than the second is possible.
|
|
Undoubtedly, however, the most important generalisation is that to several independent variables, i.e. to partial differential equations. For this really is the problem in the general case, and only in exceptional cases will it be possible to split up the disturbed partial differential equation, by the introduction of suitable variables, into separate differential equations, each only with one variable.
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|
§ 2. Several Independent Variables (Partial Differential Equation)
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|
We will represent the several independent variables in the formulae
|
|
j symbolically by the one sign x, and briefly write dx (instead of
|
|
J. .. j dx1dx2 ••• ) for an integral extending over the multiply-
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|
dimensioned domain. A notation of this type is already in use in the theory of integral equations, and has the advantage, here as there, that the structure of the formulae is not altered by the increased number of variables as such, but only by essentially new occurrences, which may be related to it.
|
|
Let therefore L[y] now signify a self-adjoint partial linear differential expression of the second order, whose explicit form we do not require to specify; and further let p(x) again be a positive function of the independent variables, which does not vanish in general. The postulation "self-adjoint" is now no longer unimportant, as the property cannot now be general1y gained by multiplication by a suitably chosen f (x), as was the case with one variable. In the particular differential expression of wave mechanics, however, this is still the case, as it arises from a variation principle.
|
|
According to these definitions or conventions, we can regard equation (2) of § I,
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|
(2)
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|
|
L[y] + Epy =0,
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|
as the formulation of the Sturm-Liouville proper value problem in the case of several variables also. Everything said there about the proper values and functions, their orthogonality, normalisation, etc., as also the whole perturbat'l:on theory there developed-in short, the whole of§ I-remains valid without change, when all the proper values are simple, if we use the abbreviated symbolism just agreed upon above. And only one thing does not remain valid, namely, that they must be simple.
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70
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|
|
WA VE MECHANICS
|
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|
|
Nevertheless, from the pure mathematical standpoint, the case when the roots are all distinct is to be regarded as the general case for several variables also, and multiplicity regarded as a special occurrence, which, it is admitted, is the rule in applications, on account of the specially simple and symmetrical structure of the differential expressions L[y] (and the "boundary conditions") which appear. Multiplicity of the proper values corresponds to degeneracy in the theory of conditioned periodic systems and is therefore especially interesting for quantum theory.
|
|
A proper value Ek is called a-fold, when equation (2), for E = Ek, possesses not one but exactly a linearly independent solutions which satisfy the boundary conditions. We will denote these by
|
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|
|
(16)
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|
|
Ukl, U1-2, . . . Uka.•
|
|
|
|
Then it is true that each of these a proper functions is orthogonal to
|
|
|
|
each of the other proper functions belonging to another proper value
|
|
|
|
(the factor p(x) being included; cf. (3)). On the contrary, these a
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|
|
functions are not in general orthogonal to one another, if we merely
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|
|
postulate that they are a linearly independent proper functions for
|
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|
|
the proper value Ek, and nothing more. For then we can equally well
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|
|
replace them by a arbitrary, linearly independent, linear aggregates
|
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|
|
(with constant coefficients) of themselves. We may express this
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|
|
otherwise, thus. The series of functions (16) is initially indefinite to
|
|
|
|
the extent of a linear transformation (with constant coefficients),
|
|
|
|
involving a non-vanishing determinant, and such a transformation
|
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|
|
destroys, in general, the mutual orthogonality.
|
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|
|
But through such a transformation this mutual orthogonality can
|
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|
|
always be brought about, and indeed in an infinite number of ways ;
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|
|
the latter property arising because orthogonal transformation does not
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|
|
destroy the mutual orthogonality. We are now accustomed to include
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|
|
this simply in normalisation, that orthogonality is secured for all
|
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|
|
proper functions, even for those which belong to the same proper
|
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|
|
value. We will assume that our uki's are already normalised in this
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|
{f way, and of course for each proper value. Then we must have
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|
|
(l7)
|
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|
|
p(x)uki(x)uk'i'(x)dx=O when (k, i) =f=(k', i')
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|
|
= I when k' =k, as well as i' =i.
|
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|
|
Each of the finite series of proper functions uki, obtained for constant k _and varying i, is then only still indefinite to this extent, that it is subject to an orthogonal transformation.
|
|
We will now discuss, first in words, without using formulae, the consequences which follow when a perturbing term is added to the differential equation (2). The addition of the perturbing term will, in general, remove the above-mentioned symmetry of the differential equation, to which the multiplicity of the proper values (or of certain of them) is due. Since, however, the proper values and functions are continuously dependent on the coefficients of the differential equation, a small perturbation causes a group of a proper values, which lie close
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|
|
QUANTISATION AND PROPER VALUES-III
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|
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|
71
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|
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|
to one another and to Ek, to enter in place of the a-fold proper value Ek. The latter is split up. Of course, if the symmetry is not wholly destroyed by the perturbation, it may happen that the splitting up is not complete and that several proper values (still partly multiple) of, in summa, equal multiplicity merely appear in the place of Ek (" partial removal of degeneracy").
|
|
As for the perturbed proper functions, those a members which belong to the a values arising from Ek must evidently also on account of continuity lie infinitely near the unperturbed functions belonging to E1c, viz. Uki; i = I, 2, 3 . . . a. Yet we must remember that the last-named series of functions, as we have established above, is indefinite to the extent of an arbitrary orthogonal transformation. One of the infinitely numerous definitions, which may be applied to the series of functions, uki ; i =], 2, 3 . . . a, will lie infinitely near the series of perturbed functions ; and if the value Ek is completely split up, it will be a quite definite one! For to the separate simple proper values, into which the value is split up, there belong proper functions which are quite uniquely defined.
|
|
This unique particular specification of the unperturbed proper functions (which may fittingly be designated as the "approximations of zero order" for the perturbed functions), which is defined by the nature of the perturbation, will naturally not generally coincide with that definition of the unperturbed functions which we chanced to adopt to begin with. Each group of the latter, belonging to a definite a-fold proper value Ek, will have first to be submitted to an orthogonal substitution, defined by the kind of perturbation, before it can serve as the starting-point, the "zero approximation", for a more exact definition of the perturbed proper functions. The defining of these orthogonal substitutions-one for each multiple proper value-is the only essentially new point that arises because of the increased number of variables, or from the appearance of multiple proper values. The defining of these substitutions forms the exact counterpart to the finding of an approximate separation system for the perturbed motion in the theory of conditioned periodic systems. As we will see immediately, the definition of the substitutions can always be given in a theoretically simple way. It requires, for each a-fold proper value, merely the principal axes transformation of a quadratic form of a (and thus of a finite number of) variables.
|
|
When the substitution has once been accomplished, the calculation of the approximations of the first order runs ahnost word for word as in § I. The sole difference is that the dash on the sigma in equation (14) must mean that in the summation all the proper functions belonging to the value Ek, i.e. all the terms whose denominators would vanish, must be left out. It may be remarked in passing that it is not at all necessary, in the calculation of first approximations, to have completed the orthogonal substitutions referred to for all multiple proper values, but it is sufficient to have done so for the value Ek, in whose splitting up we are interested. For the approximations
|
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|
72
|
|
|
|
WAVE MECHANICS
|
|
|
|
of higher order, we admittedly require them all. In all other respects, however, these higher approximations are from the beginning carried out exactly as for simple proper values.
|
|
Of course it may happen, as was mentioned above, that the value Ek, either generally or at the initial stages of the approximation, is not completely split up, and that multiplicities (" degeneracies ") still remain. This is expressed by the fact that to the substitutions already frequently mentioned there still clings a certain indefiniteness, which either always remains, or is removed step by step in the later approximations.
|
|
Let us now represent these ideas by formulae, and consider as before the perturbation caused by (4), § 1,
|
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|
|
(4)
|
|
|
|
-Ar(x)y,
|
|
|
|
i.e. we imagine the proper value problem belonging to (2) solved, and now consider the exactly corresponding problem (2'),
|
|
|
|
(2')
|
|
|
|
L[y]-,\ry + Epy =0.
|
|
|
|
We again fix our attention on a definite proper value Ek. Let (16) be a system of proper functions belonging to it, which we assume to be normalised and orthogonal to one another in the sense described above, but not yet fitted to the particular perturbation in the sense explained, because to find the substitution that leads to this fitting is precisely our chief task! In place of (5), § 1, we must now put for the perturbed quantities the following,
|
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|
a
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|
(18)
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|
|
E*lcl=E1&+A€z; u*kz(x)= L Kliuki(x)+Avi(x)
|
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|
i=l
|
|
|
|
wherein the vz(x)'s are functions, and the £i's and the Kli's are systems of constants, which are still to be defined, but which we initially do not limit in any way, although we Im.ow that the system of coefficients Kli must 1 form an orthogonal substitution. The index k should still be attached to the three types of quantity named, in order to indicate that the whole discussion refers to the kth proper value of the unperturbed problem. We refrain from carrying this out, in order to avoid the confusing accumulation of indices. The index k is to be assumed fixed in the whole of the following discussion, until the contrary is stated.
|
|
Let us select one of the perturbed proper functions and values by giving a definite value to the index l in (18), and let us substitute from (18) in the differential equation (2') and arrange in powers of ,\, Then the terms independent of ,\ disappear exactly as in § 1, because the unperturbed proper quantities satisfy equation (2),
|
|
1 It follows from the general theory that the perturbed system of functions u*1:1(:r:) mwt be orthogonal if the_ :eerturbation completely removes the degeneracy,
|
|
and may be assumed orthogonal ·a-!Mlough• that is not the case.
|
|
-, t.' ~' , : ,') l
|
|
|
|
\tUA.l'l .1..1.DA.1.J.V.l'l .ti.HJ../ .1. .l.\,V..[ .lcJ.1.\, \' .fi.LIU.c.io-.1.J..1.
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|
Iv
|
|
|
|
by hypothesis. Only terms containing the first power of .,\ remain, as we can strike out the others. Omitting a factor A, we get
|
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|
a.
|
|
|
|
(19)
|
|
|
|
L[vz] + EkPVz = L Kzi(r - Ezp )uJci,
|
|
|
|
i=l
|
|
|
|
and thus obtain again for the definition of the perturbation vz of the functions a non-homogeneous equation, to which corresponds as homogeneous equation the equation (2), with the particularvalueE = Ek, i.e. the equation satisfied by the set of functions UJci; i = 1, 2, ... a. The form of the left side of equation (19) is independent of the index l.
|
|
On the right side occur Ez and Kzt, the constants to be defined, and we are thus enabled to evaluate them, even before calculating Vz. For, in order that (19) should have a solution at all, it is necessary and sufficient that its right-hand side should be orthogonal to all the proper functions of the homogeneous equation (2) belonging to Ek. Therefore, we must have
|
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|
|
.i: Kuf(r - Ezp)u1ctu1cmdx = 0
|
|
|
|
(20)
|
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|
|
{ i=l
|
|
|
|
(m=l, 2, 3 . . . a),
|
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|
|
i.e. on account of the normalisation (17),
|
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|
|
(21)
|
|
|
|
f KzmEz = . KzifrukiUkmdx
|
|
f t=l
|
|
l (m = 1, 2, 3 ... a).
|
|
|
|
If we write, briefly, for the symmetrical matrix of constants, which can be evaluated by quadrature,
|
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|
|
(22 ) then we recognise in (21')
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|
|
{ = f ruktu,..,,,,dx Eim (i, m = I, 2, 3 . . . a),
|
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|
|
t Kzm€l =
|
|
|
|
K[iEmi
|
|
|
|
i=l {
|
|
(m=l, 2, 3 . . . a}
|
|
|
|
a system of a linear homogeneous equations for the calculation of the a constants Kzm; m = l, 2 . . . a, where the perturbation Ez of the proper value still occurs in the coefficients, and is itself unknown. However, this serves for the calculation of Ez before that of the Kzm's. For it is known that the linear homogeneous system (21 ') of equations has solutions if, and only if, its determinant vanishes. This yields the following algebraic equation of degree a for Ez :
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|
€11 - €[, €12
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' • . • €111
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=0.
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(23)
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|
€21
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' €22 - €[, • • • €21
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.. ,.J...,._,~ .J...,.&......,.v..1...1...ci..&..,ivu
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|
We see that the problem is completely identical with the transformation of the quadratic form in a variables, with coefficients Emi, to its principal axes. The " secular equation " (23) yields a roots for £t, the " reciprocal of the squares of the principal axes ", which in general are different, and on account of the symmetry of the Em/s always real. We thus get all the a perturbations of the proper values (l = 1, 2 . . . a) at the same time, and would have inferred the splitting up of an a-fold proper value into exactly a simple values, generally different, even had we not assumed it already, as fairly obvious. For each of these Ez-values, equations (21') give a system of quantities Kz;,; i = 1, 2, . . . a, and, as is known, only one (apart from a general constant factor), provided all the E/s are really different. Further, it is known that the whole system of a 2 quantities Kti forms an orthogonal system of coefficients, defining as usual, in the principal axes problem, the directions of the new co-ordinate axes with reference to the old ones. We may, and will, employ the undefined factors just mentioned to normalise the Kz/s completely as " direction cosines ", and this, as is easily seen, makes the perturbed proper functions u*ki( x) tum out normalised again, according to (18), at least in the "zero approximation" (i.e. apart from the A-terms).
|
|
If the equation (23) has multiple roots, then we have the case previously mentioned, when the perturbation does not completely remove the degeneration. The perturbed equation has then multiple proper values also and the definition of the constants Kzi becomes partially arbitrary. This has no consequence other than that (as is always the case with multiple proper values) we must and may acquiesce, even after the perturbation is applied, in a system of proper functions which in many respects is still arbitrary.
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|
The main task is accomplished with this transformation to principal axes, and we will often find it sufficient in the applications in quantum theory to define the proper values to a first and the functions to zero approximation. The evaluation of the constants Kzi and Ezi cannot be carried out always, since it depends on the solution of an algebraic equation of degree a. At the worst there are methods 1 which give the evaluation to any desired approximation by a rational process. We may thus regard these constants as known, and will now give the calculation of the functions to the first approximation, for the sake of completeness. The procedure is exactly as in§ 1.
|
|
We have to solve equation (19) and to that end we write Vz as a series of the whole set of proper functions of (2),
|
|
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|
(24)
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|
V1(x) = ~ Yl k'i'Uk'i'(x).
|
|
(k' i') ,
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|
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|
The summation is to extend with respect to k' from O to ctJ, and, for each fixed value of k', for i' varying over the finite number of proper functions which belong to Ek'• (Now, for the first time, we take account of proper functions which do not belong to the a-fold value
|
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|
1 Courant-Hilbert, chap. i. § 3. 3, p. 14.
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|
"\'.V4l., .1..lU4.1..LV.J.., I'.l..J..,.J..J .L .l.l,V.L .l.:..Ll, Y I'.l..LIV.L:.IJ-.L.L.L
|
|
Ek we are fixing our attention on.) Secondly, we develop the righthand side of (19), divided by p(x), in a series of the entire set of proper functions,
|
|
(25) wherein
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(26)
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|
* = i~ Kzifruhuk'i'dx for k' k
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1
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|
=0
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|
fork' =k
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(the last two equalities follow from (17) and (20) respectively). On substituting from (24) and (25) in (19), we get
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|
(27)
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|
L Yt, k'i'(L[uk'i'] + Ekpuk,i') = L Ct,k'i,PUk'i'•
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|
(k'i')
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|
|
(k'i')
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|
Since uk'i'.' satisfies equation (2) with E = Ek, this gives
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(28)
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|
L Yt,k'i'P(Ek-Ek,)uk'i'= l: Ct,k'i'Puk'i'•
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|
(k'i')
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|
(k'i')
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|
By equating coefficients on right and left, all the Yl,k'i''s are defined,
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|
|
with the exception of those in which k' = k. Thus
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|
(29)
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|
J Yt,lc'i'=ECz'kE'i' =E l E la.: Kti rukiuk'i'dx(fork'*k),
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|
k- le' k- le' i=l
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|
|
while those y's for which k' = k are of course not fixed by equation (19). This again corresponds to the fact that we have provisionally normalised the perturbed functions u*kt, of (18), only in the zero approximation (through the normalisation of the Kz/s), and it is easily recognised again that we have to put the who]e of the y-quantities in question equal to zero, in order to bring about the normalisation of the u*kz's even in the first approximation. By substituting from (29) in (24), and then from (24) in (18), we finally obtain for the perturbed proper
|
|
functions to a first approximation
|
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|
(30)
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|
u*kz(x)=
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|
la:.
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|
Kti( uki(x)+,\
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|
L'
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|
E u
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|
k
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|
"
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|
i
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|
'E(x
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|
)
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|
|
J ruki11,k'i'dx )
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|
i=l
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|
(k'i') k - k'
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|
(l = 1, 2, . . ., a).
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|
|
The dash on the second sigma indicates that all the terms with k' = k are to be omitted. In the application of the formula for an arbitrary k, it is to be observed that the Kz/s, as obviously also the multiplicity a of the proper value Ek, to which we have specialJy directed our attention, still depend on the index k, though this is not expressed in the symbols. Let us repeat here that the Kz/s are to be calculated as a system of solutions of equations (21'), normalised so that the sum of the squares is unity, where the coefficients of the equations are given by (22), while for the quantity Ez in (21'), one of the roots of
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IV
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|
H ..t1.. Y ~ .l.U~\.I.J..l..t1...J..'I .l\.11..J
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(23) is to be taken. This root then gives the allied perturbed proper value, from
|
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|
|
(31)
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|
|
E*kl =Ek+ llEz.
|
|
|
|
Formulae (30) and (31) are the generalisations of (14) and (15) of§ 1. It need scarcely be said that the extensions and generalisations
|
|
mentioned at- the end of § 1 can of course take effect here also. It is hardly worth the trouble to carry out these developments generally. We succeed best in any special case if we do not use ready-made formulae, but go directly by the simple fundamental principles, which have been explained, perhaps too minutely, in the present paper. I would only like to consider briefly the possibility, already mentioned at the end of § 1, that the equation (2) perhaps may lose (and indeed in the case of several variables irreparably lose), its self-adjoint character if the perturbing terms also contain derivatives of the unknown function. From general theorems we know that then the proper values of the perturbed equation no longer need to be real. We can illustrate this further. We can easily see, by carrying out the developments of this paragraph, that the elements of determinant (23) are no longer symmetrical, when the perturbing term contains derivatives. It is known that in this case the roots of equation (23) no longer require to be real.
|
|
The necessity for the expansion of certain functions in a series of proper functions, in order to arrive at the first or zero approximation of the proper values or functions, can become very inconvenient, and can at least complicate the calculation considerably in cases where an extended spectrum co-exists with the point spectrum and where the point spectrum has a limiting point (point of accumulation) at a finite distance. This is just the case in the problems appearing in the quantum theory. Fortunately it is often-perhaps alwayspossible, for the purpose of the perturbation theory, to free oneself from the generally very troublesome extended spectrum, and to develop the perturbation theory from an equation which does not possess such a spectrum, and whose proper values do not accumulate near a finite value, but grow beyond all limits with increasing index. We will become acquainted with an example in the next paragraph. Of course, this simplification is only possible when we are not interested in a proper value of the extended spectrum.
|
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|
|
II. APPLICATION TO THE STARK EFFECT
|
|
§ 3. Calculation of Frequencies by the Method which corresponds to
|
|
that of Epstein
|
|
If we add a potential energy + eFz to the wave equation (5), Part I. (p. 2), of the Kepler problem, corresponding to the influence of an electric field of strength Fin the positive z-direction, on a negative
|
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|
|
QUANTISATION AND PROPER VALUES-III
|
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|
77
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|
|
|
electron of charge e, then we obtain the following wave equation for the Stark effect of the hydrogen atom,
|
|
|
|
{32)
|
|
|
|
which forms the basis of the remainder of this paper. In § 5 we will apply the general perturbation theory of § 2 directly to this partial differential equation. Now, however, we will lighten our task by introducing space parabolic co-ordinates Ai, A2, </,, by the following equations,
|
|
(x-~cos\b
|
|
|
|
(33)
|
|
|
|
'ly=~smef,
|
|
|
|
i = ½("-1 -,\2)-
|
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|
|
.,\1 and .,\2 run from Oto infinity; the corresponding co-ordinate surfaces are the two sets of confocal paraboloids of revolution, which have the origin as focus and the positive (.,\2) or negative (,\1) z-axis respectively !:ts axes. <p runs from O to 21r, and the co-ordinate surfaces belonging to it are the set of ha.If planes limited by the z-axis. The relation of the co-ordinates is unique. For the functional determinant we get
|
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|
|
(34)
|
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|
|
o(x, y, z) - .l(' '\ ) 3(,\1, "-2, <p) - 4 "1 + "2 •
|
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|
|
The space element is thus
|
|
|
|
(35)
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|
|
dxdydz = ¼(,\1 +,\2)d,\1d,\2d<f>.
|
|
|
|
We notice, as consequences of (33),
|
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|
|
(36)
|
|
|
|
x2 + y2 = "-1"-2 ; r2 = x2 + y2 + z2 = {½(,\1 + "-2)}2.
|
|
|
|
The expression of (32) in the chosen co-ordinates gives, if we multiply by (34) 1 (to restore the self-adjoint form).
|
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|
|
l !f) !f) +f)~2t a~ ("-1 +a~ ("-2 +¼(I
|
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|
|
(32')
|
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|
|
1
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|
|
1
|
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|
2
|
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|
|
2
|
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|
|
1 2 't'
|
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|
|
+ h212r2m[E(,\1 +,\2) + 2e2 -½eF(,\12 -,\22)],f, =0.
|
|
|
|
Here we can again take-and this is the why and wherefore of all " methods " of solving linear partial differential equations-the
|
|
function rp as the product of three functions, thus,
|
|
|
|
(37)
|
|
|
|
if,= A1A2<l>,
|
|
|
|
1 So far as the actual details of the analysis are concerned, the simplest way to get (32'), or, in general, to get the wave equation for any special co-ordinates, is to transform not the wave equation itself, but the corresponding variation problem (cf. Part I. p. 12), and thus to obtain the wave equation afresh as an Eulerian variation problem. We are thus spared the troublesome evaluation of the 8econd derivatives. Cf. Coura.nt-Hilbert, chap. iv. § 7, p. 193.
|
|
|
|
78
|
|
|
|
WAVE MECHANICS
|
|
|
|
each of which depends on only one co-ordinate. For these functions we get the ordinary differential equations
|
|
cJ2<1> cJef,2 = - n2<I>
|
|
|
|
a(\ (38)
|
|
|
|
OAi
|
|
|
|
/\1 calAA1)
|
|
|
|
+
|
|
|
|
2-7,;T,2zm(
|
|
|
|
-
|
|
|
|
i F\ 2
|
|
2e /\1
|
|
|
|
+
|
|
|
|
E\ /\1
|
|
|
|
+
|
|
|
|
e 2
|
|
|
|
-
|
|
|
|
a
|
|
fJ
|
|
|
|
-
|
|
|
|
Sn72Th22m
|
|
|
|
Al)A1
|
|
|
|
-0
|
|
-,
|
|
|
|
1
|
|
|
|
1
|
|
|
|
a ( oA2
|
|
|
|
A2aoAA:)
|
|
|
|
+
|
|
|
|
~ 27T2m\( ½eFA22
|
|
|
|
+ EA2
|
|
|
|
+
|
|
|
|
e2
|
|
|
|
+ f1
|
|
|
|
-
|
|
|
|
n2h2 81T2m
|
|
|
|
1) A2 A2
|
|
|
|
=
|
|
|
|
0,
|
|
|
|
wherein n and f3 are two further "proper value-like" constant,s of integration (in addition to E), still to be defined. By the choice of symbol for the first of these, we have taken into account the fact that the first of equations (38) makes it take integral values, if <I> and
|
|
|
|
~: are to be continuous and single-valued functions of the azimuth ef,.
|
|
|
|
We then have
|
|
|
|
(39)
|
|
|
|
Cl> =Sill n<p
|
|
cos
|
|
|
|
and it is evidently sufficient if we do not consider negative values of n. Thus
|
|
|
|
(40)
|
|
|
|
n=O, 1, 2, 3 . . . .
|
|
|
|
In the symbol used for the second constant /3, we follow Sommerfeld
|
|
|
|
(Atombau, 4th edit., p. 821) in order to make comparison easier.
|
|
|
|
(Similarly, below, with A, B, 0, D.) We treat the last two equations
|
|
|
|
of (38} together, in the form
|
|
|
|
(41}
|
|
|
|
a~(t~J) + (ng2 +At+ 2B + i)A =0,
|
|
|
|
where
|
|
|
|
(42}
|
|
|
|
D1 }
|
|
D2
|
|
|
|
=
|
|
|
|
1T2meF =t= h2 '
|
|
|
|
A= 21T2mE h2 '
|
|
|
|
B1 } B
|
|
|
|
7T2m =v(
|
|
|
|
e 2
|
|
|
|
a =FtJ),
|
|
|
|
0= -4n2,
|
|
|
|
2
|
|
|
|
and the upper sign is valid for A =A1, g=A.1, and the lower one for
|
|
A =A2, g =A2. (Unfortunately, we have to write g instead of the more appropriate A, to avoid confusion with the perturbation para-
|
|
|
|
meter A of the general theory, §§ 1 and 2.}
|
|
|
|
If we omit initially in (41) the Stark effect term D[2, which we
|
|
|
|
conceive as a perturbing term (limiting case for vanishing field), then
|
|
|
|
this equation has the same general structure as equation (7) of Part I.,
|
|
|
|
and the domain is also the same, from Oto a:;;. The discussion is almost
|
|
|
|
the same, word for word, and shows that non-vanishing solutions,
|
|
|
|
which, with their derivatives, are continuous and remain finite within
|
|
|
|
the domain, only exist if either A > 0 (extended spectrum, correspond-
|
|
|
|
ing to hyperbolic orbits} or
|
|
|
|
(43)
|
|
|
|
~ B -~ =k + ½; k=O, l, 2, . . .
|
|
|
|
+
|
|
|
|
QUANTISATION AND PROPER VALUES-III
|
|
|
|
79
|
|
|
|
If we apply this to the last two equations of (38) and distinguish the two k-values by suffixes 1 and 2, we obtain
|
|
|
|
v'"=""A(k1 + ½+ V-0) =B1
|
|
|
|
+
|
|
|
|
+
|
|
|
|
(44)
|
|
|
|
{ '\,I-A(k2 + ½+ V-0) =B2.
|
|
|
|
+
|
|
|
|
+
|
|
|
|
By addition, squaring and use of (42) we find
|
|
|
|
41r4m2e4
|
|
|
|
21r 2me4
|
|
|
|
(45)
|
|
|
|
A= - h4l2 and E= - ~ ·
|
|
|
|
These are the well - known Balmer-Bohr elliptic levels, where as principal quantum number enters
|
|
|
|
(46)
|
|
|
|
l = k1 + k2 + n + I.
|
|
|
|
We get the discrete term spectrum and the allied proper functions
|
|
|
|
in a way simpler than that indicated, if we apply results already
|
|
|
|
known in mathematical literature as follows. We transform first the
|
|
|
|
dependent variable A in (41) by putting
|
|
|
|
(47)
|
|
t and then the independent by putting
|
|
|
|
(48)
|
|
|
|
2tv-=-7r =17.
|
|
|
|
We find for u as a function of 17 the equation
|
|
|
|
(41')
|
|
|
|
d2u d172
|
|
|
|
n +
|
|
|
|
+ 17
|
|
|
|
1
|
|
|
|
du d17
|
|
|
|
+
|
|
|
|
(
|
|
|
|
D (2yCA_)a17
|
|
|
|
_
|
|
|
|
1
|
|
4
|
|
|
|
__!!____!) _
|
|
+ '\,/-A 17 u-0.
|
|
|
|
+
|
|
This equation is very intimately connected with the polynomials named after Laguerre. In the mathematical appendix, it will be
|
|
X
|
|
shown that the product of e- 2 and the nth derivative of the (n + k)th
|
|
Laguerre polynomial satisfies the differential equation
|
|
|
|
(103)
|
|
|
|
y,,+n~+y1 + , ( - ! + (k +n ~+ 1x)1y) =0,
|
|
|
|
and that, for a fixed n, the functions named form the complete system of proper functions of the equation just written, when k runs through all non-negative integral values. Thus it follows that, for vanishing D, equation (41') possesses the proper functions
|
|
|
|
(49)
|
|
|
|
and the proper values
|
|
|
|
(50)
|
|
|
|
V
|
|
|
|
B_ A
|
|
|
|
=
|
|
|
|
n+I - 2-
|
|
|
|
+
|
|
|
|
k
|
|
|
|
(k = o, I, 2 . . .)
|
|
|
|
+
|
|
-and no others! (See the mathematical appendix concerning the remarkable loss of the extended spectrum caused by the apparently inoffensive transformation (48) ; by this loss the development of the perturbation theory is made much easier.)
|
|
|
|
80
|
|
|
|
WAVE MECHANICS
|
|
|
|
We have now to calculate the perturbation of the proper values
|
|
(50) from the general theory of § 1, caused by including the D-term in (41'). The equation becomes self-adjoint if we multiply by 11"+ 1. The density function p( x) of the general theory thus becomes 11n· As
|
|
perturbation function r(x) appears
|
|
|
|
(51)
|
|
|
|
- h(2 ,.,,r•.
|
|
|
|
+
|
|
|
|
(We formally put the perturbation parameter A= 1 ; if we desired,
|
|
|
|
we could identify D or F with it.) Now formula (7') gives, for the
|
|
|
|
perturbation of the lcth proper value,
|
|
|
|
D llJ11n+2e-11[L:+k(11)]2d11
|
|
|
|
(52)
|
|
|
|
E&= - ( 2 p ) • (' 71ne-'[L:H(11)]'d17 •
|
|
|
|
. 0
|
|
|
|
For the integral in the denominator, which merely provides for the normalisation, formula (115) of the appendix gives the value
|
|
|
|
(53)
|
|
|
|
[(n +k) !]3 k!
|
|
|
|
while the integral in the numerator is evaluated in the same place, as
|
|
|
|
(54)
|
|
|
|
[(
|
|
|
|
n
|
|
|
|
+k) 1]3 k!. (n
|
|
|
|
2+
|
|
|
|
6n
|
|
|
|
k
|
|
|
|
+
|
|
|
|
6
|
|
|
|
k2+
|
|
|
|
6
|
|
|
|
k
|
|
|
|
+
|
|
|
|
3
|
|
|
|
n
|
|
|
|
+
|
|
|
|
2
|
|
|
|
)
|
|
|
|
.
|
|
|
|
Consequently
|
|
|
|
(55)
|
|
|
|
Ek=
|
|
|
|
-
|
|
|
|
(
|
|
|
|
\iD=A)/n2
|
|
2
|
|
|
|
+
|
|
|
|
6nk+6k2+6k+3n+2).
|
|
|
|
+
|
|
|
|
The condition for the kth perturbed proper value of equation (41')
|
|
|
|
and therefore, naturally, also for the kth discrete proper value of the
|
|
|
|
original equation (41) runs therefore
|
|
|
|
(56)
|
|
|
|
-v"-B=- = A
|
|
|
|
-n++kl +E 2
|
|
|
|
k
|
|
|
|
+
|
|
|
|
( Ek is retained meantime for brevity).
|
|
|
|
This result is applied twice, namely, to the last two equations
|
|
|
|
of (38) by substituting the two systems (42) of values of the constants
|
|
|
|
A, B, C, D ; and it is to be observed that n is the same number
|
|
|
|
in the two cases, while the two k-values are to be distinguished by
|
|
|
|
the suffixes 1 and 2, as above. First we have
|
|
|
|
v"B=1 A
|
|
|
|
=
|
|
|
|
n+l -2-
|
|
|
|
+
|
|
|
|
k1
|
|
|
|
+
|
|
|
|
Ek}
|
|
|
|
(57)
|
|
|
|
+
|
|
|
|
\lUAl~ J.J.~AJ.J.Vl~ A1~1.J r .n.vr ~.n. V 11..LU~C-111.
|
|
|
|
OJ.
|
|
|
|
whence comes (58)
|
|
|
|
(applying abbreviation (46) for the principal quantum number). In the approximation we are aiming at we may P-xpand with respect
|
|
to the small quantities Ek and get
|
|
|
|
(59)
|
|
|
|
1 A
|
|
|
|
=
|
|
|
|
-
|
|
|
|
(B1
|
|
|
|
+l2B2)
|
|
|
|
2 [
|
|
|
|
1 -
|
|
|
|
2( Ek1
|
|
|
|
+ Ek2)]•
|
|
|
|
Further, in the calculation of these small quantities, we may use the approximate value (45) for A in (55). We thus obtain, noticing the two D values, by (42),
|
|
|
|
Eki = + 6=:e5(n2 + 6nk1 + 6k12 + 6k1 + 3n + 2)
|
|
|
|
(60)
|
|
|
|
( Ek2 =
|
|
|
|
-
|
|
|
|
Fh'l3 641r'm2e5(n
|
|
|
|
2
|
|
|
|
+
|
|
|
|
6nk2
|
|
|
|
+
|
|
|
|
6k22
|
|
|
|
+
|
|
|
|
6k2
|
|
|
|
+
|
|
|
|
3n
|
|
|
|
+
|
|
|
|
2).
|
|
|
|
Addition gives, after an easy reduction,
|
|
|
|
(61)
|
|
|
|
Ek1
|
|
|
|
+
|
|
|
|
Ek2
|
|
|
|
=
|
|
|
|
3Fh4l4(k1 - k2)
|
|
321r'm2e5
|
|
|
|
'
|
|
|
|
If we substitute this, and the values of A, B1, and B2 from (42) in (59), we get, after reduction,
|
|
|
|
(62)
|
|
|
|
E = _ 21r2me' _ ~ h2Fl(k2 - k1). h2l2 8 1r2m.e
|
|
|
|
This is our provisional conclusion; it is the well-known formula of Epstein for the term values in the Stark effect of the hydrogen spectrum.
|
|
k1 and k2 correspond fully to the parabolic quantum numbers ; they are capable of taking the value zero. Also the integer n, which has evidently to do with the equatorial quantum number, may from (40) take the value zero. However, from (46) the sum of these three numbers must still be increased by unity in order to yield the principal
|
|
quantum number. Thus (n + I) and not n corresponds to the equatorial quantum number. The value zero for the latter is thus automatwally excluded by wave mechanics, just as by Heisenberg's mechanics.1 There is simply no proper function, i.e. no state of vibration, which corresponds to such a meridional orbit. This important and gratifying circumstance was already brought to light in Part I. in counting the constants, and also afterwards in§ 2 of Part I. in connection with the azimuthal quantum number, through the non-existence of states of vibration corresponding to pendulum orbits ; its full meaning, how-
|
|
ever, only fully dawned on me through the remarks of the two authors just quoted.
|
|
|
|
1 W. Pauli, jun., Ztschr. f. Phys. 36, p. 336, 1926; N. Bohr, Die Naturw. I,
|
|
1926.
|
|
|
|
VV AV .n, lU.n.linA1>; llii::i
|
|
|
|
For later application, let us note the system of proper functions
|
|
|
|
of equation (32) or (32') in "zero approximation", which belongs to the
|
|
|
|
proper values (62). It is obtained from statement (37), from con-
|
|
|
|
clusions (39) and (49), and from consideration of transformations (47)
|
|
|
|
and (48) and of the approximate value (45) of A. For brevity, let us
|
|
|
|
call a0 the " radius of the first hydrogen orbit". Then we get
|
|
|
|
(63)
|
|
|
|
I
|
|
|
|
h2
|
|
|
|
2lyCA =411'2me2 = ao.
|
|
|
|
The proper functions (not yet normalised!) then read
|
|
|
|
(64)
|
|
|
|
i - (~)Ln (~) ./. - ,\ ~,\
|
|
"fnk1k2 - 1 2 e
|
|
|
|
,
|
|
|
|
\
|
|
|
|
2
|
|
|
|
!
|
|
|
|
,O \
|
|
|
|
z
|
|
|
|
Ln n
|
|
|
|
+
|
|
|
|
k
|
|
|
|
1
|
|
|
|
lao
|
|
|
|
n+ka lao
|
|
|
|
scions
|
|
|
|
,J. n"f•
|
|
|
|
They belong to the proper values (62), where l has the meaning (46). To each non-negative integral trio of values n, k1, k2 belong
|
|
( on account of the double symbol ~~~) two proper functions or one,
|
|
|
|
according as n > 0 or n = 0.
|
|
|
|
§ 4. Attempt to calculate the Intensities and Polarisations of the
|
|
Stark Effect Patterns
|
|
|
|
I have lately shown 1 that from the proper functions we can calculate by differentiation and quadrature the elements of the matrices, which are allied in Heisenberg's mechanics to functions of the generalised position- and momentum-co-ordinates. For example, for the (rr')th element of the matrix, which according to Heisenberg belongs to the generalised co-ordinate q itself, we find
|
|
|
|
f q" = qp(x)r/,r(x)rf,,,(x)dx
|
|
|
|
(65)
|
|
|
|
l
|
|
|
|
! .{! -f r. p(x)[,f,,(x)]"dx p(x)[,f,,(x)]2dx
|
|
|
|
Here, for our case, the separate ind.ices each deputise for a trio of indices n, k1, k2, and further, x represents the three co-ordinates
|
|
r, 0, ef,. p(x) is the density function; in our case the quantity (34).
|
|
(We may compare the self-adjoint equation (32') with the general form (2)). The "denominator" (. . .)-l in (65) must be put in because our system (64) of functions is not yet normalised.
|
|
According to Heisenberg, 2 now, if q means a rectangular Cartesian co-ordinate, then the square of the matrix element (65) is to be a measure of the "probability of transition" from the rth state to the r'th, or, more accurately, a measure of the intensity of that part of the radiation, bound up with this transition, which is polarised in the q-direction. Starting from this, I have shown in the above paper that if we make
|
|
|
|
1 Preceding paper of this collection. 2 W. Heisenberg, Ztschr. f. Phys. 33, p. 879, 1925; M. Born and P. Jordan, Ztschr. f. Phys. 34, pp. 867, 886, 1925.
|
|
|
|
(JU A.N TliSATlU .N A.NV .t'.l:W.t'~.K VAL U ~i:;-111
|
|
|
|
ts::S
|
|
|
|
certain simple assumptions as to the electrodynamical meaning of if,, the " mechanical field scalar ", then the matrix element in question is susceptible of a very simple physical interpretation in wave mechanics, namely, actually: component of the amplitude of the periodically oscillating electric moment of the atom. The word component is to be taken in a double sense: (I) component in the q-direction, i.e. in the spatial direction in question, and (2) only the part of this spatial component which changes in a time-sinusoidal manner with exactly the frequency
|
|
of the emitted light, IEr -Er· l/h. (It is a question then of a kind of
|
|
Fourier analysis: not in harmonic frequencies, but in the actual frequencies of emission.) However, the idea of wave mechanics is not that of a sudden transition from one state of vibration to another, but according to it, the partial moment concerned-as I will briefly name it-arises from the simultaneous existence of the two proper vibra-
|
|
tions, and lasts just as long as both are excited together. Moreover, the above assertion that the qrr''s are proportional to the
|
|
partial moments is more accurately phrased thus. The ratio of, e.g., q" to q"w is equal to the ratio of the partial moments which arise when the proper function 1/Jr and the proper functions 1/lr' and if,rw are stimulated, the first with any strength whatever and the last two with
|
|
strengths equal to one another-i.e. corresponding to normalisation. To calculate the ratio of the intensities, the q-quotient must first be squared and then multiplied by the ratio of the fourth powers of the emission frequencies. The latter, however, has no part in the intensity ratio of the Stark effect components, for there we only compare intensities of lines which have practically the same frequency.
|
|
The known selection and polarisation rules for Stark effect com-
|
|
ponents can be obtained, almost without calculation, from the integrals in the numerator of (65) and from the form of the proper functions in (64). They follow from the vanishing or non-vanishing of the
|
|
integral with respect to ¢,. We obtain the components whose
|
|
electric vector vibrates parallel to the field, i.e. to the z-direction, by replacing the q in (65) by z from (33). The expression for z, i.e.
|
|
½(A1 - ,\2), does not contain the azimuth ¢,. Thus we see at once
|
|
from (64) that a non-vanishing result after integration with respect
|
|
to ¢, can only arise if we combine proper functions whose n's are
|
|
equal, and thus whose equatorial quantum numbers are equal, being in fact equal to n + I. For the components which vibrate perpendicular to the field, we must put q equal to x or equal to y
|
|
(cf. equation (33)). Here cos ¢, or sin ¢, enters, and we see almost
|
|
as easily as before, that the n-values of the two combined proper functions must differ exactly by unity, if the integration with respect
|
|
to ¢, is to yield a non-vanishing result. Hence the known selection
|
|
and polarisation rules are proved. Further, it should be recalled again that we do not require to exclude any n-value after additional reflection, as was necessary in the older theory in order to agree with experience. Our n is smaller by I than the equatorial quantum number, and right from the beginning cannot take negative values
|
|
|
|
84
|
|
|
|
WAVE MECHA.N !US
|
|
|
|
(quite the same state of affairs exists, we know, in Heisenberg's theory}. 1
|
|
The numerical evaluation of the integrals with respect to .-\1 and .-\2 which appear in (65) is exceptionally tedious, especially for those of the numerator. The same apparatus for calculating comes into play as served already in the evaluation of (52), only the matter is somewhat more detailed because the two (generalised) Laguerre polynomials, whose product is to be integrated, have not the same argument. By good luck, in the Balmer lines, which interest us principally, one of
|
|
the two polynomials L:+k, namely that relating to the doubJy quantised state, is either a constant or is a linear function of its argument. The method of calculation is described more fully in the mathematical appendix. The following tables and diagrams give the results for the first four Balmer lines, in comparison with the known measurements and estimates of intensity, made by Stark 2 for a field strength of about 100,000 volts per centimetre. The first column indicates the state of polarisation, the second gives the combination of the terms in the usual manner of description, i.e. in our symbols : of the two trios of numbers {k1, k2, n + I) the first trio refers to the higher quantised state and the second to the doubly quantised state. The third column, with the heading d, gives the term decomposition in multiples of 3h2 F/87r2me, (see equation (62)). The next column gives the intensities observed by Stark, and O there signifies not observed. The question mark was put by Stark at such lines as clash either with irrelevant lines or with possible "ghosts" and thus cannot be guaranteed. On account of the unequal weakening of the two states of polarisation in the spectrograph, according to Stark his results for the 11 and for the ...L components of vibration are not directly comparable with one another. Finally, the last column gives the results of our calculation in relative numbers, which are comparable for the collective components ( II and ...L) of one line, e.g. of H11, but not for those of Ha with HfJ, etc. These relative numbers are reduced to their smallest integral values, i.e. the numbers in each of the four tables are prime to each other.
|
|
1 W. Pauli, jun., Ztschr. f. Physik, 36, p. 336, 1926.
|
|
2 J. Stark, Ann. d. Phys. 48, p. 193, 1915.
|
|
|
|
[TABLES
|
|
|
|
QUANTISATION AND PROPER VALUES-III
|
|
|
|
85
|
|
|
|
INTENSITIES IN THE STARK EFFECT OF THE BALMER LINES
|
|
TABLE 1
|
|
|
|
Polarisation.
|
|
II
|
|
l.
|
|
|
|
Combination.
|
|
(lll) (Oll) (102) (002) (201) (101) (201) (Oll)
|
|
|
|
~
|
|
|
|
Observed Intensity. Calculated Intensity.
|
|
|
|
2
|
|
|
|
1
|
|
|
|
3
|
|
|
|
1-1
|
|
|
|
4
|
|
|
|
1·2
|
|
|
|
8
|
|
|
|
0
|
|
|
|
729 2304 1681
|
|
1
|
|
|
|
Sum: 4715
|
|
|
|
(003) (002) (lll) (002)
|
|
|
|
} 0
|
|
0
|
|
|
|
2·6
|
|
|
|
{
|
|
|
|
4608 882
|
|
|
|
(102) (101)
|
|
|
|
1
|
|
|
|
1
|
|
|
|
1936
|
|
|
|
(102) (Oll)
|
|
|
|
5
|
|
|
|
0
|
|
|
|
16
|
|
|
|
(201) (002)
|
|
|
|
6
|
|
|
|
0
|
|
|
|
18
|
|
|
|
• Undlsplaced components halved.
|
|
|
|
Sum*: 4715
|
|
|
|
Polarisation.
|
|
II
|
|
l.
|
|
|
|
Combination.
|
|
(112) (002) (2ll) (101)
|
|
-
|
|
(2ll) (Oll) (202) (002) (301) (101)
|
|
-
|
|
(301) (011)
|
|
-
|
|
(ll2) (Oll) (103) (002) (211) (002) (202) (101)
|
|
-
|
|
(202) (011) (301) (002)
|
|
|
|
TABLE 2
|
|
H.a
|
|
|
|
~
|
|
|
|
Observed Intensity. Calculated Intensity.
|
|
|
|
0
|
|
|
|
1-4
|
|
|
|
0
|
|
|
|
2
|
|
|
|
1·2
|
|
|
|
9
|
|
|
|
(4)
|
|
|
|
1
|
|
|
|
0
|
|
|
|
6
|
|
|
|
4·8
|
|
|
|
81
|
|
|
|
8
|
|
|
|
9·1
|
|
|
|
384
|
|
|
|
10
|
|
|
|
1H5
|
|
|
|
361
|
|
|
|
(12)
|
|
|
|
1
|
|
|
|
0
|
|
|
|
14
|
|
|
|
0
|
|
|
|
1
|
|
|
|
Sum: 836
|
|
|
|
(0)
|
|
|
|
1-4
|
|
|
|
0
|
|
|
|
2
|
|
|
|
3·3
|
|
|
|
72
|
|
|
|
} 4
|
|
4
|
|
|
|
12·6
|
|
|
|
{
|
|
|
|
384 72
|
|
|
|
6
|
|
|
|
9·7
|
|
|
|
294
|
|
|
|
(8)
|
|
|
|
1·3
|
|
|
|
0
|
|
|
|
10
|
|
|
|
1-1?
|
|
|
|
6
|
|
|
|
12
|
|
|
|
1 ?
|
|
|
|
8
|
|
|
|
Sum: 836
|
|
|
|
86
|
|
|
|
WAVE MECHANICS
|
|
|
|
INTENSITIES IN THE STARK EFFECT OF THE BALMER LINES
|
|
TABLE 3
|
|
Hy
|
|
|
|
Polarisation.
|
|
ll
|
|
|
|
Combination.
|
|
(221) (011) (212) (002) (311) (101) (311) (011) (302) (002) (401) (101) (401) (011)
|
|
|
|
A
|
|
|
|
Observed Intensity. Calculated Intensity.
|
|
|
|
2
|
|
|
|
1-6
|
|
|
|
5
|
|
|
|
1-5
|
|
|
|
8
|
|
|
|
1
|
|
|
|
12
|
|
|
|
2·0
|
|
|
|
15
|
|
|
|
7·2
|
|
|
|
18
|
|
|
|
10·8
|
|
|
|
22
|
|
|
|
l ?
|
|
|
|
15 625 19 200
|
|
l 521 16 641 115 200 131 769
|
|
729
|
|
|
|
Sum: 300 685
|
|
|
|
(113) (002) (221) (002)
|
|
|
|
} 0
|
|
0
|
|
|
|
7·2
|
|
|
|
{
|
|
|
|
115 200 26 450
|
|
|
|
(212) (101)
|
|
|
|
3
|
|
|
|
3·2
|
|
|
|
46 128
|
|
|
|
(212) (011)
|
|
|
|
7
|
|
|
|
1-2
|
|
|
|
5 808
|
|
|
|
...L
|
|
|
|
(203) (002) (311) (002)
|
|
|
|
} 10
|
|
10
|
|
|
|
4·3
|
|
|
|
{
|
|
|
|
76 800 11 250
|
|
|
|
(302) (101)
|
|
|
|
13
|
|
|
|
6·1
|
|
|
|
83 232
|
|
|
|
(302) (011)
|
|
|
|
17
|
|
|
|
1-1
|
|
|
|
2 592
|
|
|
|
(401) (002)
|
|
|
|
20
|
|
|
|
l
|
|
|
|
4 050
|
|
|
|
• Undisplaced components halved.
|
|
|
|
Sum : * 300 685
|
|
|
|
Polarisation. II
|
|
...L
|
|
|
|
Combination.
|
|
(222) (002) (321) (101) (321) (011) (312) (002) (411) (101) (411) (011) (402) (002) (501) (101) (501) (011)
|
|
(222) (0ll) (213) (002) (321) (002) (312) (101) (312) (011) (303) (002) (411) (002) (402) (101) (402) (011) (501) (002)
|
|
|
|
TABLE 4
|
|
Hs
|
|
|
|
A
|
|
|
|
Observed Intensity. Calculated Intensity.
|
|
|
|
0
|
|
|
|
0
|
|
|
|
0
|
|
|
|
4
|
|
|
|
l
|
|
|
|
8
|
|
|
|
8
|
|
|
|
1·2
|
|
|
|
32
|
|
|
|
12
|
|
|
|
1-5
|
|
|
|
72
|
|
|
|
16
|
|
|
|
1·2
|
|
|
|
18
|
|
|
|
20
|
|
|
|
1-1
|
|
|
|
18
|
|
|
|
24
|
|
|
|
2·8
|
|
|
|
180
|
|
|
|
28
|
|
|
|
7·2
|
|
|
|
242
|
|
|
|
32
|
|
|
|
l ?
|
|
|
|
2
|
|
|
|
Sum: 572
|
|
|
|
2
|
|
|
|
1·3
|
|
|
|
36
|
|
|
|
} 6
|
|
6
|
|
|
|
3·2
|
|
|
|
{
|
|
|
|
162 36
|
|
|
|
IO
|
|
|
|
2·1
|
|
|
|
98
|
|
|
|
14
|
|
|
|
l
|
|
|
|
2
|
|
|
|
} 18
|
|
18
|
|
|
|
2·0
|
|
|
|
{
|
|
|
|
90 9
|
|
|
|
22
|
|
|
|
2·4
|
|
|
|
125
|
|
|
|
26
|
|
|
|
1·3
|
|
|
|
5
|
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30
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1 ?
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9
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Sum: 572
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QUANTISATION AND PROPER VALUES-III
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87
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In the diagrams it is to be noticed that, on account of the huge
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differences in the theoretical intensities, some theoretical intensities
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e.rp.
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111
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~ J 2
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111
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2 3 I/
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e:,:p.
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I I
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1 0 1
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theor.
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0
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R
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III I II
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~J 2
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2 J ~
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FIG. 1,-Ha. II-components.
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fheor:
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s .I 6 5
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I I
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• I
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10 1
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5 6
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FJG. 2.-Ha. 1. -components
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exp.
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exp.
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I
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~
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ro
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a
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6
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I
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~
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z I
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o '
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I
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2
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I
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~
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6
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B
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ro
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I
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~
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theor.
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I I
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I
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0
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10 8 6
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2
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2
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6 8 10 1fl.
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Fm. 3.-H/J II-components.
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fheor.
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. '
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12 10 6 'I- 2
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2 4 6
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10 12
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Fm. 4.-HfJ 1. -components.
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88
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WAVE MECHANICS
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cannot be truly represented to scale, as they are much too small. These are indicated by small circles.
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A consideration of the diagrams shows that the agreement is to]erably good for almost aJI the strong components, and taken all over it is somewhat better than for the values deduced from correspondence considerations.1 Thus, for example, is removed one of the most serious contradictions which arose, in that the correspondence principle gave the ratio of the intensities of the two strong ..L-components of HfJ, for .!l = 4 and 6, inversely and indeed very much out, in fact
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exp.
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I l III III
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?
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I
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22 18 15 12 8 5 2 2 S 8 12 15 18 22
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f/Jeor.
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0
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22 18 15 12
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' I
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85 2
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0
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25 8
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FIG. 5.-H1 II-components.
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12 ,., 18
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0
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22
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as almost 1 : 2, while experiment requires about 5 : 4. A similar thing occurs with the mean (.!l =0) ..L-components of H1, which decidedly preponderate experimentally, but are given as far too weak by the correspondence principle. In our diagrams also, it is admitted that such "reciprocities" between the intensity ratios of intense components demanded by theory and by experiment are not entirely wanting. The theoretically most intense II-component (.:'.l = 3) of Ha is furthest out ; by experiment, it should lie be.tween its neighbours
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in intensity. And the two strongest 11- components of Hfj and two
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..L-components (.:'.l = 10, 13) of H1 are given "reciprocally " by the
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1 H. A. K.ramers, Diiniache Akademie (8), iii. 3, p. 333 d Bt.JJ.., 1919.
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