COLLECTED PAPERS ON WAVE MECHANICS BY E. SCHRODINGER PROFESSOR OF' THEORETICAL PHYSICS AT THE UNIVERSITY OF BERLIN TOGETHER WITH HIS FOUR LECTURES ON WAVE MECHANICS CHELSEA PUBLISHING COMPANY NEW YORK, N. Y. THIRD (AUGMENTED) EDITION 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 THIS THIRD ENGLISH EDITION INCORPORATES THE FULL TEXT OF A WORK ORIGINALLY PUBLISHED AS A SEPARATE WORK: 'FouR LECTURES ON WAVE MECHANICS.' THIS WAS ORIGINALLY PUBLISHED AT GLASGOW IN 1928 AND REPRINTED AT GLASGOW IN 1929. THE PRESENT WORK IS PRINTED ON 'LONG-LIFE' ACID-FREE PAPER. COPYRIGHT @, 1982 BY CHELSEA PUBLISHING COMPANY LIBRARY OF CONGRESS CATALOG CARD NUMBER 80-70108 INTERNATIONAL STANDARD BOOK NUMBER 0-8284-1302-9 PRINTED IN THE UNITED STATES OF AMERICA Contents PAPERS PAQJI:. ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART I . . . . . . . . . . . . 1 2. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART II . . . . . . . . . . . . 13 3. THE CONTINUOUS TRANSITION FROM MICRO- TO MACRO-MECHANICS 41 4. ON THE REL\TION BETWEEN THE QUANTUM MECHANICS OF HEISENBERG, BORN, AND JORDAN, AND THAT OF SCHR0DINGER....... . 45 5. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART III........... . 62 6. QUANTISATION AS A PROBLEM OF PROPER VALUES. PART IV 102 7. THE COMPTON EFFECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8. THE ENERGY-MOMENTUM THEOREM FOR MATERIAL WAVES . . . . . . . . . . . . . . 130 9. THE EXCHANGE OF ENERGY ACCORDING TO WAVE MECHANICS . . . . . . . . . . 137 LECTURES 1. DERIVATION OF THE FUNDAMENTAL IDEA OF WAVE MECHANICS FROM HAMILTON'S ANALOGY BETWEEN ORDINARY MECHANICS AND GEOMETRICAL OPTICS ....................................... _ 155 2. ORDINARY MECHANICS ONLY AN APPROXIMATION, WHICH NO LONGER HOLDS FOR VERY SMALL SYSTEMS ......................... _ 160 3. BOHR'S STATIONARY ENERGY-LEVELS DERIVED AS THE FREQUENCIES OF PROPER VIBRATIONS OF THE WAVES............................. _ 163 V VI CONTENTS 4. ROUGH DESCRIPTION OF THE WAVE-SYSTEMS IN THE HYDROGEN ATOM. DEGENERACY. PERTURBATION ............................ 168 5. THE PHYSICAL MEANING OF THE WAVE FUNCTION. EXPLANATION OF THE SELECTION RULES AND OF THE RULES FOR THE POLARIZATION OF SPECTRAL LINES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6. DERIVATION OF THE WAVE EQUATION (PROPERLY SPEAKING) 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 STATE OF THE ATOM PRODUCED BY INCIDENT RADIATION WHOSE FREQUENCY COINCIDES, OR NEARLY COINCIDES, WITH A NATURAL EMISSION FREQUENCY .................................... 185 10. EXTENSION OF WAVE MECHANICS TO SYSTEMS OTHER THAN A SINGLE MASS-POINT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,, . . . . . . . . . . . 189 11. EXAMPLES: THE OSCILLATOR, THE ROTATOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 12. CORRECTION FOR MOTION OF THE NUCLEUS IN THE HYDROGEN 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 Preface to the First (German) Edition 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. 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. E. SCHRODINGER. ZURICH, November 1926. Vll Publishers' Note 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. 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. 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. viii Abstract (The references are to pages.) 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 ix X WAVE MECHANICS (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). 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 Atlt::;T.KAt..J'l' XI (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. 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. 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 (pp. 104-117) ; in all this the electrodynamic hypothesis about ip is 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.). 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. 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 XU WAVE M..1£UHA.NllJI:; 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. 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 1 Cf. also a paper by W. Gordon on the Compton Effect, Ztschr. J. Phys. 40, p. 117, 1926. ABSTRACT xiii 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. WA VE MECHANICS Quantisation as a Problem of Proper Values (Part I) (Annalen der Physik (4), vol. 79, 1926) § 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. The usual form of the latter is connected with the Hamilton-Jacobi differential equation, (I) A solution of this equation is sought such as can be represented as the sum of functions, each being a function of one only of the independent variables q. Here we now put for Sa new unknown tp such that it will appear as a poduct, of related functions of the single co-ordinates, i.e. we put (2) S =K logy,. The constant K must be introduced from considerations of dimensions ; it has those of action. Hence we get (I') H (q, Klf,oayq,) =E. 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') can always be transformed so as to become a quadratic form (of y, and its first derivatives) equated to zero. (For the one-electron problem I VV AV .l!.i lVl.l!.i\.i.tl.Al~ 1\.iO this holds even when mass-variation is not neglected.) We now seek a function y,, such that for any arbitrary variation of it the integra] of the said quadratic form, taken over the whole co-ordinate space,1 is stationary, y, being everywhere real, single-valued, finite, and con- tinuously differentiable up to the second order. The quantum conditions are repl,aced by this variation -problem. First, we will take for H the Hamilton function for Keplerian motion, and show that ip can be so chosen for all positive, but only for a discrete set of negative values of E. That is, the above variation problem has a discrete and a continuous spectrum of proper values. The discrete spectrum corresponds to the Balmer terms and the continuous to the energies of the hyperbolic orbits. For numerical agreement K must have the value h/21r. The choice of co-ordinates in the formation of the variational equa- tions being arbitrary, let us take rectangular Cartesians. Then (l') becomes in our case (l") ( ao-yx,)2 + (a-oyy,) 2 + (o-oi'z1)2 - -K2m-2 (E +e-r2) y,2 = 0 · ' e = charge, m = mass of an· electron, r2 = x2 + y2 + z2• Our variation problem then reads (:tr (a:J -~~(E (3) 8J =8fffdxdydz[ +(:r + + t)f•] =0, the integral being taken over all space. From this we find in the usual way J Jff ~":\E (4) ½8J = df8y,:t-- dx.dy dz 8i/J[,;2y, + + ~)y,] = 0. Therefore we must have, firstly, (5) V2YJ + 2Km2 (E +-e;.2) 'P =0, and secondly, (6) df is an element of the infinite closed surface over which the integral is taken. (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.) The solution of (5) can be effected,for example, in polar co-ordinates, r, B, cf,, if y, be written as the product of three functions, each only of r, of B, or of ATlUl~ Al~V rn,vr .r..n Y.11..1....1u.J.!lo-.L tends to infinity in a definite way, but also to give an account of this behaviour for one defim'.te solution, which is always a much more difficult task. We shall at first exclude the case where a1 and a2 are real integers. When this occurs, it occurs for both quantities simultaneously, and when, and only when, m,e2 (15) Kv-2mE=a real integer. -2mE + Therefore we assume that (15) is not fulfilled. The behaviour of the totality of solutions when r tends to infinity 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 (16) lim ezr = 0, z ➔ ao i.e. the real part of zr is to become negative and infinite. In this way condition (13) is satisfied. In the one case let z make a circuit once round the point c (solution U and in the other, round c (solution U ), ). 1 1 2 2 Now for very large real positive values of r, these two solutions are represented asymptotically (in the sense used by Poincare) by (17) u1,_,ec1rr-o.1( - l)0. 1(e21Tio., -I)r(a1){C1 -C2)°"2 - 1, { U2,.._,ec2rr-o.2( - I)o.2(e21rio.2 - l)r(a2)(c2 - C1)0.1 -1, in which we are content to take the first term of the asymptotic series of integral negative powers of r. 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. 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. 6 WAVE MEC.tiA.NlC8 2. E < 0. In this case the possibility (15) is not eo ipso excluded, yet we wiil maintain that exclusion provisionally. Then by (14") and (17), 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 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 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 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 U. (12) can be developed in a series of positive powers of r, which converges, at all events, for r sufficiently small, and since it satisfies equation (7'), it must coincide with the series for U. Therefore U is 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 additively combined from the two paths, considered above, which 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. Our integral function U, which alone of the solutions of (7') is considered for our problem, is therefore not finite for r large, on the above hypothesis. Reserving meanwhile the question of completeness, i.e. the proving that our treatment allows us to find all the linearly independent solutions of the problem, then we may state: For negative values of E which do not satisfy condition (15) our variation problem has no solution. We have now only to investigate that discrete set of negative 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- 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 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 and the integration effected completely in terms of known functions, so that the behaviour of the solutions can be fully investigated. Let (15') Vme2 l ; l = 1, 2, 3, 4 . . . K -2mE Then from (14") we have (14"') a1 -1 =l+n, a2 -1 = -l+n. Two cases have to be distinguished : l 6= n and l > n. (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 to fulfil condition (13). A third characteristic point here is at infinity (negative and real). Every path between two of these three points yields a solution, and of these three solutions there are two linearly in- ~U.tt...J.., .1.J.IJ.tt..-1.-1.V..1.., .n....1..,~ .L .L1,V.L .L...1..1.1, , ..._.._~'-'...._...., .._ 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. 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 solution x of the original equation (7). (The multiplying constant is arbitrary.) We find (18) - ( v-2mE), x-f r K , f ( x ) -x n e -xl-n- k~o 1(~ -2x)k (l+n l-n - I - ) k. 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 : 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). Taking into account the constants in the surface harmonic (known to be 2n + I in number), it is further found that : 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. 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. 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. 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, and therefore olji/or only tends to zero on an infinite sphere as 1/r2• 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 u VY 11. V ~ lVl.l~.A.iilAl.~ ll.iO 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. § 2. Condition (15) yields me4 (19) -Ez = 2K2z2. 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 (20) from which comes K=~, 27T (19') 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. 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 (21) v -K K 2l h2l al 2mE me2 = 41T2me2 = l' where az = the semi-axis of the lth elliptic orbit. ( The equations follow from (19) plus the known relation Ei= ;~} 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. QUANTISATION AND PROP.EH. VALUES-J. § 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. 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). 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. 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. 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 1 L. de Broglie, Ann. de Physique (10) 3, p. 22, 1925. (Theses, Paris, 1924.) 1 Phy8ik. Zt,'lchr. 27, p. 95, 1926. WAVE MECHANICS 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. 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. 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. 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 \:lU.l"1..J.., .1..LUI1..L.LV..Ll 4'..&...._,....., -L ........ _ ..... _ _ ,.. . --- - 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. 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. 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. Addition at the proof correction on 28.2.1926. 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 " n 1-1. V ~ ll'l~vlllU'il\..ii:) (23) stationary, while fulfilling the normalising, accessory condition (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. It is to be remarked that in the quantity a 2 of (14") we have essentially the well-known Sommerfeld expression - vB'A + v1f. (Cf. Atombau, 4th (German) ed., p. 775.) Physical Institute of the University of Zurich. (Received January 27, 1926.) Quantisation as a Problem of Proper Values (Part II) (Annalen der Physik (4), vo1. 79, 1926) § I. The Hamiltonian Analogy between Mechanics and Optics 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 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 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 wave equation and the Hamilton-Jacobi equation. If is not actually the action 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. 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, 13 , , ..n. f i:, .i.U..Dvlll\.1'41 ~i:', Let us consider the general problem of conservative systems m classical mechanics. The Hamilton-Jacobi equation runs (1) aa,w: +T(qk, aaqwk I\1+ V(qk)=o. 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 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 (2) and obtain W = - Et + S(q1c), (I') E is an arbitrary integration constant and signifies, as is known, the 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. Equation (l') can now be very simply expressed if we make use of the method of Heinrich Hertz. It becomes, like all geometrical 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. Let T be the kinetic energy as function of the ve]ocities rjk, not of the momenta as above, and let us put for the line element (3) ds2 = 2T(qk, rjk)dt2• The right-hand side now contains dt only externally and represents (since 2 + ~ i f , =0, which has the solution (30) if,= sin cos [ 'V{8~ ;2EA.. ]. Here the argument must be an integral multiple of 12} + [µ.a' -µ.b'(r - r0)2]f = 0. 1 Physik. Ztschr. 27, p. 95, I 926. 38 WA V..l:!i 1Vl~lil1.Al'41L,i:, 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 (44) X =rf, and as new independent variable (45) p =r -r0• The substitution gives (46) a2x ap2+ [ µ , , a - µ , b' p 2 - n(n+l) (ro+p)2 ] x -_O . 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 function by terms of the relative order of magnitude of !!. . This is seen, To if we develop thus : (47 ) n(n + 1) (ro + p) 2 _ - n(n + ro 2 1)(1 _ 2p ro + 3p2_ ro 2 + ' • ' ) ' 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. (48 ) , n(n+l) a( p =p- ' 1o b' J-L + 3n(n + ro 4 1)) then equation (46) takes the form (46') ;}; + (a - bp'2 + [~])x = o, where we have put (49) J , a=µ,a - n(n + r2 0 l )(1 n(n + l) - r04µ , b ' + 3 n ( n + l ) ) lb=µ/,'+ 3n(;.: I)_ The symbol[~] in (46') represents terms which are small compared with the retained term of the order of i_. 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 [i] for equation (46'), if we neglect the term and bear in mind the ro. (.lUAl~Tli::,A'.llUl~ Al'UJ r .n,vr.n,.n, V11.LU.Co:,-u ,J,7 order of magnitude of molecular constants, is indeed small compared 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 (50) E = -n'-(-n--+--l)-h'-2- = n-(-n-'+--l)-h-2 167r4vo2µ,2ro4 l6114vo2A2 instead, we can easily derive the following energy steps, n(n+l)h2( e ) 2l+l . ~ (51) E:E,+ { B1r•A . 1 =l + 3• + -2 -hv0v I +3e (n-0, I, 2 . . . , l-0, 1, 2 . . .), where (52) A= µ,r02 is still written for the moment of inertia. In the language of classical mechanics, e: is the square of the ratio 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. 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. 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 40 WA V..I!: M.l!JUHA.N lUS 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''. 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. Zurich, Physical Institute of the University. (Received February 23, 1926.) The Continuous Transition from Microto Macro-Mechanics (Die NaJ,urwissenschaften, 28, pp. 664-666, 1926) 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 (1) 2m(dq)2 dt, + 2 2 2-»,,r,2 1T Vo ""'J. ' when we put, instead of the displacement q, the dimensionless variable (2) x=q. 21rJm;~, we get if, as the superposition of the following proper vibrations : 5 -f Y,n = e H n(x)e21rivnt (3) { ( v n 2n+l =-2- v0 ; n=O, 1, 2, 3 . . . ). The Hn's are the polynomials 6 named after Hermite. If they are 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). 41 42 WAVE MECHANICS x• multiplied by e-2 and the "normalising factor" (2nn !)-½ they are called Hermite's orthogonal functions. They represent therefore the amplitudes of the proper vibrations. 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. 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 Fro. 1.-The first five proper vibrations of the Planck oscillator according to undulatory mechanics. Outside of the region - 3 :s;; x.:s;; + 3 represented here, all five functions approach the x-axis in monotonic fashion. 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: (4) 'P = i ~ (~)n'Pn =eTTi11ot (1e21ri11ot)n _!__e-f Hn(x). n=O 2- n ! n=O 2 n ! Thus the normalised proper vibrations (see above) are taken with the coefficients (5) 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 A2 (6) n = 2- · 1 zn/n ! has, as function of n,for large values of z, a single extremely high and relatively very sharp maximum at n=z. By taking square roots and with z=A 2/2, we get the series of numbers (5). MICRO- AND MACRO-M~CHANlCS The summation of the series (4) is made possible by the following identity 1 in x and s : (7) --n ( ) k~~J n=O n S 1n e • ~ 2 n x =e - s 1 + 2 s z ~ -2-. Thus (8) Now we take, as is provided for, the real part of the right-hand side and after a short calculation obtain (9) if,= e~•-t(x-.A cos 2"v0t)2 cos [ 1rv0t + (A sin 21rv0t). ( X - : cos 21rv0t)]. 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 (10) x = A cos 21rv0t. The breadth of the hump is of the order of magnitude unity and therefore very small compared with A, by hypothesis. According to (.10), the hump oscillates under exactly the same law as would operate in the usual mechanics for a particle having (1) as its energy function. The amplitude in terms of xis A, and thus in terms of q is (11) a=A11i..f h. 21r" mv0 Ordinary mechanics gives for the energy of a particle of mass m, which oscillates with this amplitude and with frequency v0, (12) 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 1 Courant-Hilbert, loc. cit. eqn. (58). 44 WAVE MECHANICS 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 -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. 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. 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 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 46 WA V.h: M~CHAl'HC8 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 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). MATRIX MECHANICS AND WAVE MECHANICS 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. § 2. The Co-ordination of an Operator and of a Matrix with a Wellarranged Function-symbol and the Establishment of the Product Rule 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 differential operators obey in the domain of the single set of n variables, 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 a · Actua11y t h e operator u~qz 1s exc ha ngea bl e w·ith ~O-, uqm where m ~1zs arbitrary, but with qm only, if m-:t=l. The operator, obtained by interchange and subtraction when m = l, viz. (1) o-aqqz z -qzoaq-z, 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. 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 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,#· • • We wish to express this as a "well-arranged i function-symbol" and relate it to the following operator, 1 Or " well-ordered." 48 WAVE MECHANICS .., wherein, somewhat more generally than in the preliminary survey, x!, Pr is not replaced by } simply, but by and K stands for a vqr Vlf.r universal constant. As an abbreviation for the operator arising out of the well-arranged function F, I have introduced the symbol [F, • ] in passing (i.e. only for the purpose of the present proof). The function (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 [G F, u] will denote the function u after the operator of F has first been used on it, and then the operator of G; or, what is defined to be the same, when the operator of GF has been used. Of course this is not generally the same as [FG, u]. Now we connect a matrix with a well-arranged function, like F, by means of its operator (3) and of an arbitrary complete orthogonal system having for its domain the whole of q-space. It is done as follows. For brevity we will simply write x for the group of f variables qi, q2, ••• qn, as is usual in the theory of Integral Equations, and write dx for an integral extending over the whole of q-space. The functions (4) u1(x)Vp(x}, u2(x)Vp(x}, u3(x)Vp(x} . . . ad inf. are now to form a complete orthogonal system, normalised to 1. {f Let, therefore, in every case {5) p(x)ui(x)uk(x)dx=O for i-4:k =l for i=k. 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. By the operator (3) we now relate the following matrix, (6) Fkl=jp(x)uk(x)[F, ui(x)]dx, 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 MAT.lU.X l\.l.1£<.J.tlA..l'HC~ A.NlJ WAV ~ lVrnlJtlAl''HlJi:; 4~ 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 (7) the matrix corresponding. We wish to form the product matrix (FG)km = ~Fkl(]lm, l Before writing it, let us transform the expression (6) for Fkl as follows. By a series of integrations by parts, the operator [F, • ] is "revolved" from the function ui(x) to the function p(x)uk(x). By the expression "revolve " (instead of, say, " push ") I wish to convey that the sequence of the operations reverses itself exactly thereby. The boundary integrals, which come in as "by-products", are to disappear (see above). The "revolved " operator, including the change of sign that accompanies an odd number of differentiations, will be denoted by [F, • ]. For example, from (3) comes a2 a (3') [F, • ] = ( - l)T . . . K 2u~qsh"uq(,,,q, 1 . . qn)K-u;qs,-,, a a a aa g(ql . . . qn)K3 qt q, q,, f(q1 . . . qn), where ,- =number of differentiations. By applying this symbol, we have (6') 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} = j[F, p(x)uk(x)][G, um(x)]dx. The last equation is simply the so-called "relation of completeness " 1 More briefly: Fk1 is the kth "development coefficient" of the operator used on the function u 1• WA V.hi MEUHA.N !CS of our orthogonal system,1 applied to the " development coefficients ,, of the functions 1 [G, Um(x)] and p(xlF, p(x)uk(x)]. 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 (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. § 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. That is Heisenberg's "quantum relation" if we put (12) K= h , 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 { J2 = ft,2~ -idx = I. J f ... f As above, dx stands for dq1 •.• dqn; dp-l is the reciprocal 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 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.) If we indicate the derivative of T with respect to that argument, 1 Equations (23) and (24) of Part I. 56 WAVE MECHANICS which originally was Pk, by the suffix Pk, we obtain, as the result of the variation, O=½(8J1 - E8J2) (30) ~J =J{ -::, f :t.)J dp-•Tp,(q~ + (V(q•) - E)dp-trp}a,f,dx; the Eulerian variation equation thus runs : (31) f oifi)} a{ ( sh1r22L\i aqk L\p - •TP1; qk, oqk - V(qk)t/J + Ey, = o. It is not difficult to see that this equation has the form of (21) if we remember our law connecting the operators, and consider (32) T(qk, p1c) = ½Lp1cTP (% p1c) le J; the Eulerian equation for homogeneous functions, applied to the quadratic form T. In actual fact, if we detach the operator from the left side of (31), with the proper value term Ey, removed, and replace in it ~ 27T -1 u~qa k by P>, then according to (32) we obtain the negatively taken Hamilton function (28). Thus the process of variation has given quite automatically a uniquely defined " symmetricalisation " of the operator, which makes it self-adjoint (except possibly for a common factor) and makes it invariant for point transformations, and which I would like to maintain, as long as there are no definite reasons for the appear2.nce under the integrals (29) of the additional factor, already 1 mentioned as possible, and for a definite form of the latter. Hence the solution of the whole system of matrix equations of Heisenberg, Born, and Jordan is reduced to the natural boundary value problem of a linear partial differential equation. If we have solved the boundary value problem, then by the use of (6) we can calculate by differentiations and quadratures every matrix element we are interested in. As an illustration of what is to be understood by the natural boundary value problem, i.e. by the natural boundary conditions at the natural boundary of configuration space, we may refer to the worked examples.2 It invariably turns out that the natural infinitely distant boundary forms a singularity of the differential equation and only allows of the one boundary condition-" remaining finite ". This seems to be a general characteristic of those micro-mechanical prob- lems with which the theory in the first place is meant to deal. If the domain of the position co-ordinates is artificially limited (example : a molecule in a "vessel"), then an essential allowance must be made for this limitation by the introduction of suitable potential energies in 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. MATRIX MECHANICS AND WAVE MECHANICS 57 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. § 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. 58 WAVE MECHANICS 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 (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, (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 MATRIX MECHANICS AND WAVE MECHANICS 59 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, . We notice, as consequences of (33), (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). l !f) !f) +f)~2t a~ ("-1 +a~ ("-2 +¼(I (32') 1 1 2 2 1 2 't' + 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, 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 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 and ~: are to be continuous and single-valued functions of the azimuth ef,. We then have (39) Cl> =Sill n

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 30 1 ? 9 Sum: 572 QUANTISATION AND PROPER VALUES-III 87 In the diagrams it is to be noticed that, on account of the huge differences in the theoretical intensities, some theoretical intensities e.rp. 111 ~ J 2 111 2 3 I/ e:,:p. I I 1 0 1 theor. 0 R III I II ~J 2 2 J ~ FIG. 1,-Ha. II-components. fheor: s .I 6 5 I I • I 10 1 5 6 FJG. 2.-Ha. 1. -components exp. exp. I ~ ro a 6 I ~ z I o ' I 2 I ~ 6 B ro I ~ theor. I I I 0 10 8 6 2 2 6 8 10 1fl. Fm. 3.-H/J II-components. fheor. . ' 12 10 6 'I- 2 2 4 6 10 12 Fm. 4.-HfJ 1. -components. 88 WAVE MECHANICS cannot be truly represented to scale, as they are much too small. These are indicated by small circles. 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 exp. I l III III ? I 22 18 15 12 8 5 2 2 S 8 12 15 18 22 f/Jeor. 0 22 18 15 12 ' I 85 2 0 25 8 FIG. 5.-H1 II-components. 12 ,., 18 0 22 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 in intensity. And the two strongest 11- components of Hfj and two ..L-components (.:'.l = 10, 13) of H1 are given "reciprocally " by the 1 H. A. K.ramers, Diiniache Akademie (8), iii. 3, p. 333 d Bt.JJ.., 1919.