mm 'm BURLINGTON Four Lectures on Wave Mechanics BLACKIE & SON LIMITED 50 Old Bailey, London 17 Stanhope Street, Glasgow BLACKIE & SON (INDIA) LIMITED Warwick House, Fort Street, Bombay BLACKIE & SON (CANADA) LIMITED 1 1 18 Bay Street, Toronto Four Lectures on Wave Mechanics Delivered at the Royal Institution, London, on 5th, 7th, i2th, and 14th March, 1928 BY Dr. ERWIN SCHRODINGER Professor of Theoretical Physics in the University of Berlin BLACKIE & SON LIMITED LONDON AND GLASGOW 1928 C33 Printed in Great Britain by Blackie Gf Son, Ltd., Glasgow T)edicated to the memory of Fritz Hasenohrl m 13^ T . Contents FIRST LECTURE Page 1 Derivation of the fundamental idea of wave mechanics from Hamilton's analogy between ordinary me- chanics and geometrical optics - - - - i 2. Ordinary mechanics only an approximation, which no longer holds for very small systems - - -6 3. Bohr's stationary energy-levels derived as the fre- quencies of proper vibrations of the waves - -9 SECOND LECTURE 4. Rough description of the wave-systems in the hydrogen atom. Degeneracy. Perturbation - - - 14 5. The physical meaning of the wave function. Explana- tion of the selection rules and of the rules for the polarization of spectral lines - - - - 16 6. Derivation of the wave equation (properly speaking) which contains the time - - - - - 22 7. An atom as perturbed by an alternating electric field - 23 THIRD LECTURE 8. Theory of secondary radiation and dispersion - - 27 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 - - - 31 viii CONTENTS Page 10. Extension of wave mechanics to systems other than a single mass-point - - - - - - 35 11. Examples: the oscillator, the rotator - - - - 38 FOURTH LECTURE 12. Correction for motion of the nucleus in the hydrogen atom 43 13. Perturbation of an arbitrary system - - - - 45 14. Interaction between two arbitrary systems - - 49 15. The physical meaning of the generalized c];-function - 51 Four Lectures on Wave Mechanics FIRST LECTURE 1. Derivation of the fundamental idea of wave mechanics from Hamilton's analogy between ordinary mechanics and geometrical optics. When a mass-point in moves in a conservative field of force, described by the potential energy V(x^y, z)^ then, if you let it start from a given A point with a given velocity, i.e. with a given energy E^ you will be able to get it into another arbitrarily chosen B point by suitably " aiming ", i.e. by letting it start in a quite definitely chosen direc- tion. There is in general one definite dynamical orbit A B which leads from to zvith a given energy. This orbit possesses the property that hC2Tdt=^0, JA ... (1) and is defined by this property (Hamilton's principle in the form given to it by Maupertuis). Here T means the 2 WAVE MECHANICS kinetic energy of the mass-point, and the equation means: A B consider the manifold of all orbits leading from to and subject to the law of conservation of energy = V {T -{- E); among them the actual dynamical orbit is distinguished by the fact that, for it and for all in- finitely adjacent orbits of the manifold, the / has the same value up to small quantities of the second order (the words *' infinitely adjacent" being taken to define w = the first order of smallness). Calling ds -j the velocity of the mass-point, we have 2T=mw'' = m (j\ ^ = 2{E-V) = ~ ^2m{E-V), by means of which equation (1) can be transformed into s}^^2m{E- V)ds = 0. . . (2) This form has the advantage that the variational principle is applied to a purely geometrical integral, which does not contain the time-variable, and further, that the condition of constant energy is automatically taken care of. Hamilton found it useful to compare equation (2) with Fermafs principle, which tells us that in an optically non-homogeneous medium the actual light rays, i.e. the tracks along which energy is propagated, are determined by the " law of minimum time " (as it is usually called). Let fig. 1 now refer to an optical medium of arbitrary non-homogeneity, e.g. the earth's atmosphere; then, if you have a searchlight at A, furnishing a well-defined beam, it will in general be possible to illuminate an B arbitrarily chosen point by suitably aiming at it with the searchlight. There is one definite light-path leading FIRST LECTURE 3 A from to B, which obeys, and is uniquely defined by, the law C^ ds \ii=' (^) Here ds, as before, means the element of the path, and u is the velocity of light, a function of the co-ordinates X, y, z. The two laws contained in equations (2) and (3) respectively become identical, if we postulate that =, u= ^ , ... (4) where C must be independent of x, y, z but may depend on E. Thus we have made a mental picture of an optical medium, in which the manifold of possible light-rays coincides with the manifold of dynamical orbits of a m £ mass-point moving with given energy in a field of force V{x,y, z). The fact that 11, the velocity of light, de- pends not only on the co-ordinates but also on £", the total energy of the mass-point, is of the utmost importance. This fact enables us to push the analogy a step farther E by picturing the dependence on as dispersion, i.e. as a dependence on frequency. For this purpose we must attribute to our light-rays a definite frequency v, de- We pending on E. will (arbitrarily) put E=hv (5) {h being Planck's constant), without dwelling much on this assumption, which is very suggestive to modern physicists. Then this non-homogeneous and dispersive medium provides in its rays a picture of all the dynamical orbits of our particle. Now we can proceed a stage farther, putting the question: can we make a small 4 WAVE MECHANICS " point-like " light-signal move exactly like our masspoint? (Hitherto we have only secured the geometrical identity of orbits, quite neglecting the question of time- rate.) At first sight this seems impossible, since the velocity of the mass-point, a' = ^V2m(£-F), ... (6) is (along the path, i.e. with constant E) inversely pro- C portional to the light-velocity u (see equation (4); depends on E only). But we must remember that ii is of course the ordinary phase-YtXocity , whereas a small light-signal moves with the so-called group-velocity, say g, which is given by g dv \u/' or, in our case, following equation (5), by ' g dME i) <'> ^ We will try to make g w. The only means we have at our disposal for this purpose is a suitable choice of E C, the arbitrary function of that appeared in equation = (4). From (4), (6), and (7), the postulate g w becomes _^ /EV2m{E- V) \ dE\ C / hence ^2m{E-V) djE^ {V2m{E-V)); (^-l) ^2m(E-V) V is constant with respect to E. Since contains the E C co-ordinates and must be a function of onlv, this — FIRST LECTURE 5 relation can obviously be secured in a general way only by making the first factor vanish. Hence ^-1 = 0, or C=E, which gives equation (4) the special form ^_" E ^2m{E-Vy (8) This assumption about phase-velocity is the only one which will secure absolute coincidence between the dynamical laws of motion of the mass-point and the optical laws of motion of light-signals in our imagined light-propagation. It is worth while mentioning that, according to (8), ^^_energy_ ^g,^ momentum There is still one arbitrariness in the definition of ?/, E viz.: may obviously be changed by an arbitrary additive constant, if the same constant is added to V{x, jy, z). This arbitrariness cannot be overcome in the non- relativistic treatment and we are not going to deal with the relativistic one in the present lectures. Now the fundamental idea of wave-mechanics is the following. The phenomenon, of which we believed we had given an adequate description in the old mechanics by describing the motion of a mass-point, i.e. by giving its co-ordinates x, y, z as functions of the time variable ty — is to be described correctly according to the new ideas by describing a definite wave-motion, which takes place among waves of the type considered, i.e. of the definite frequency and velocity (and hence of the definite wavelength) which we ascribed to what we called *' light " in 6 WAVE MECHANICS the preceding. The mathematical description of a wave- motion will be furnished not by a limited number of functions of the one variable t, but by a continuous manifold, so to speak, of such functions, viz. by a func- tion (or possibly by several functions) of Xy y, z, and t. These functions will be subject to a partial differential equation, viz. to some sort of wave equation. The statement that what really happens is correctly described by describing a wave-motion does not neces- sarily mean exactly the same thing as: what really exists We is the wave-motion. shall see later on that in general- izing to an arbitrary mechanical system we are led to describe what really happens in such a system by a wave-motion in the generalized space of its co-ordinates (^-space). Though the latter has quite a definite physical meaning, it cannot very well be said to " exist "; hence " a wave-motion in this space cannot be said to *' exist in the ordinary sense of the word either. It is merely an adequate mathematical description of what happens. It may be that also in the case of one single mass-point, with which we are now dealing, the wave-motion must not be taken to " exist " m.too literal a sense, although the configuration space happens to coincide with ordinary space in this particularly simple case. 2. Ordinary mechanics only an approximation, which no longer holds for very small systems. In replacing the ordinary mechanical description by a wave-mechanical description our object is to obtain a theory which comprises both ordinary mechanical phenomena, in which quantum conditions play no appreciable part, and, on the other hand, typical quantum phenomena. The hope of reaching this object resides in the following FIRST LECTURE 7 m analogy. Hamilton's wave-picture, worked out the way discussed above, contains something that corresponds to ordinary mechanics, viz. the rays correspond to the mechanical paths, and signals move like mass-points. But the description of a wave-motion in terms of rays is merely an approximation (called " geometrical optics " in the case of light-waves). It only holds if the structure of the wave phenomenon that we happen to be dealing with is coarse compared with the wave-length, and as long as we are only interested in its " coarse structure ". The detailed fine structure of a wave phenomenon can never be revealed by a treatment in terms of rays (" geometrical optics "), and there always exist wave-phenomena which are altogether so minute that the ray-method is. of no use and furnishes no information whatever. Hence in replacing ordinary mechanics by wave mechanics we may hope on the one hand to retain ordinary mechanics as an approximation which is valid for the coarse " macro-mechanical " phenomena, and on the other hand to get an explanation of those minute " micromechanical " phenomena (motion of the electrons in the atom), about which ordinary mechanics was quite unable to give any information. At least it was unable to do so without making very artificial accessory assump- tions, which really formed a much more important part of the theory than the mechanical treatment itself.* * To give an example: the actual application of the rules for quantization to the several-electron problem was, strange to say, not hindered by the fact that nobody in the world ever knew how We to enunciate them for a non-conditionally periodic system! simply took the problem of several bodies to be conditionally periodic, though it was perfectly well known that it was not. This shows, I think, that ordinary mechanics was not made use of in a very serious manner, otherwise the said application would have been as impossible as the application of penal law to the motion of the planets. ) 8 WAVE MECHANICS The step which leads from ordinary mechanics to wave mechanics is an advance similar in kind to Huygens' We theory of light, which replaced Newton's theory. might form the symbolic proportion: Ordinary mechanics : Wave mechanics = Geometrical optics : Undulatory optics. Typical quantum phenomena are analogous to typical wave phenomena like diffraction and interference. For the conception of this analogy it is of considerable importance that the failure of ordinary mechanics does We occur in dealing with very tiny systems. can im- mediately control the order of magnitude at which a complete failure is to be expected, and we shall find that it is exactly the right one. The wave-length, say A, of our waves is (see equations (5) and (8) A = = = A ^^ V ^ ^2m{E— V) mw' (9)^ ' ' ^ i.e. Planck's constant divided by the momentum of the Now mass-point. take, for the sake of simplicity, a cir- cular orbit of the hydrogen-model, of radius «, but not necessarily a " quantized " one. Then we have by ordinary mechanics (without applying quantum rules): — mwa = n h , where n is any real positive number (which for Bohr's quantized circles w^ould be 1, 2, 3 . . . ; the occurrence of h in the latter equation is for the moment only a con- venient way of expressing the order of magnitude). Combining the last two equations, we get A_ 277 an ( D 929 ) FIRST LECTURE 9 Now in order that we may be justified in the appli- cation of ordinary mechanics it is necessary that the dimensions of the path calculated in this way should turn out to be large compared with the wave-length. This is seen to be the case as long as the '' quantum number " w is large compared with unity. As n becomes smaller and smaller, the ratio of A to « becomes less and A less favourable. complete failure of ordinary me- chanics is to be expected precisely in the region w^here we actually meet with it, viz. where n is of the order of unity, as it would be for orbits of the normal size of an atom (10~^ cm.). 3. Bohr's stationary energy-levels derived as the frequencies of proper vibrations of the waves. Let us now consider the wave-mechanical treatment of a case which is inaccessible to ordinary mechanics; say, to fix our ideas, the wave-mechanical treatment of what in ordinary mechanics is called the motion of the electron in the hydrogen atom. In what way are we to attack this problem? Well, in very much the same way as we would attack the problem of finding the possible movements (vibra- tions) of an elastic body. Only, in the latter case the problem is complicated by the existence of two types of waves, longitudinal and transverse. To avoid this complication, let us consider an elastic fluid contained in a given enclosure. For the pressure, p, say, we should have a wave equation V^/'-^J = (10) u being the constant velocity of propagation of longi- tudinal waves, the only waves possible in the case of a (D929) 2 —, 10 WAVE MECHANICS We fluid. should have to try to find the most general solution of this partial differential equation that satisfies certain boundary conditions at the surface of the vessel. The standard way of solving is to try which gives for ip the equation ^V v^^ + = o, . . . (10') i/j being subject to the same boundary conditions as p. We then meet with the well-known fact that a regular solution ip satisfying the equation and the boundary conditions cannot be obtained for all values of the co- efficient of ip, i.e. for all frequencies v, but only for an infinite set of discrete frequencies v^, Vo, Vg, . . . , j^/^, .. . which are called the characteristic or proper frequencies (Eigenfrequenzen) of the problem or of the body. Call — ^k the solution (ordinarily unique apart from a multi- plying constant) that belongs to Vk, then since the equation and the boundary conditions are homogeneous k will, with arbitrary constants Ck, Ok, be a more general solution and will indeed be the general solution, if the set of quantities {ipky i^k) is complete. (For physical appli- cations we shall of course have to use the real part of the expression (H).) In the case of the waves which are to replace in our thought the motion of the electron, there must also be some quantity p, subject to a wave equation like equation (10), though we cannot yet tell the physical meaning of p. Let us put this question aside for the moment. FIRST LECTURE ii In equation (10) we shall have to put (see above) u= ^ , (8) This is not a constant; it depends (1) on E, that is, essen- tially on the frequency v {= E/h); (2) on the co-ordinates X, y^ z, which are contained in the potential energy V. These are the two complications as compared with the simple case of a vibrating fluid body considered above. Neither of them is serious. By the first, the dependence on £", we are restricted in that we can apply the wave equation only to a function p whose dependence on the time is given by \rnEt. p -- e h whence = p p— 4:7T^E^ p (12) We need not mind that, since it is precisely the same assumption (Ansatz) as would be made in any case in the standard method of solution. Substituting from (12) and (8) in (10) and replacing the p letter by i/j (to remind us that now, just as before, we are investigating a function of the co-ordinates only), we obtain VV-f^^2'"(^-n'A = 0. . . (13) We now see that the second complication (the depen- dence of u on Vy i.e. on the co-ordinates) merely results in a somewhat more interesting form of equation (13) as compared with (10'), the quantity multiplying ip being no longer a constant, but depending on the co-ordinates. This was really to be expected, since an equation that is to embody the mechanical problem cannot very well help 12 WAVE MECHANICS A containing the potential energy of the problem. sim- m plification the problem of the " mechanical " waves (as compared with the fluid problem) consists in the absence of boundary conditions. 1 thought the latter simplification fatal when I first attacked these questions. Being insufficiently versed in mathematics, 1 could not imagine how proper vibration frequencies could appear without boundary conditions. Later on I recognized that the more complicated form of the coefficients (i.e. the appearance of F(.t, y^ z) ) takes charge, so to speak, of what is ordinarily brought about by boundary conditions, namely, the selection of definite values of E. I cannot enter into this rather lengthy mathematical discussion here, nor into the detailed process of finding the solutions, though the method is practically the same as in ordinary vibration problems, namely: introducing an appropriate set of co-ordinates (e.g. spherical or elliptical, according to the form of the function V) and putting j/f equal to a product of functions, each of which contains one co-ordinate only. I will state the result straightforwardly for the case of the hydrogen atom. Here we have to put F=-- + const., . . . (14) r being the distance from the nucleus. Then it is found that not for all, but only for the following values of E, is it possible to find regular, one-valued, and finite solu- tions ip: = - -^^^ = (A) En const. w ; 1, 2, 3, 4 . . . ] > E (B) const. J ^^^,^ FIRST LECTURE ij The constant is the same as in (14) and is (in non-relativistic wave mechanics) meaningless, except that we cannot very well give it the value which is usually adopted for the sake of simplicity, viz. zero. For then all the values (A) would become negative. And a negative frequency, if it means anything at all, means the same as the positive frequency of the same absolute value. Then it would be mysterious why all positive frequencies should be allowed, but only a discrete set of negative ones. But the question of this constant is of no importance here. You see that our differential equation automatically selects as the allowed £'-values (A) the energy-levels of the elliptic orbits quantized according to Bohr's theory; (B) all energy-levels belonging to hyperbolic orbits. This is very remarkable. It shows that, whatever the waves may mean physically, the theory furnishes a method ot quantization which is absolutely free from arbitrary postulates that this or that quantity must be an integer. Just to give an idea how the integers occur here: if e.g. (/) is an azimuthal angle and the wave amplitude turns m out to contain a factor cos m, p= -eip'^= -eI.ZckCk'ipk^k'e-'''^^'^-'^'^'+'^-'^^'\ (18) k k' where e means the absolute electronic charge. Integrating this over the whole space and making use of equations (16) and (17), we find for the total charge k which shows that we shall have to postulate k in order to make the total charge equal to the electronic charge (which we feel inclined to do). It was said before that and ?/f, hence p, is practically confined to a very small region of a few Angstrom units. — Since the wave-lengths of the light-radiations v^ Vk' are very large compared with this region, it is well known that the radiation of the fluctuating density p will be very nearly the same as that of an electric dipole whose electric moment has (e.g.) the ^-component = M^ \ \ \ zpdxdy dz (and similarly formed x- and jy- components). Calcu- SECOND LECTURE 19 M^ lating from (18), we find after an easy reduction k - -2i:ck Ck' auk' cos [2 77 {v^ v^) t-\-dk- 0,] . {k,k') (19) Here akk' is an abbreviation for the following constant: = akk' e^\jz^kh'dxdydz, . . (20) and S means a sum over all the pairs {k, k'). Hence the {k,k') squares of these integrals (and the corresponding in- tegrals relating to the x- and ^-directions) determine the — intensity of emitted light of frequency v;^ The Vy^' . | | intensity is not determined by them alone; the amplitude- constants Ck also come into play, of course. But this is quite satisfactory. For the integrals akk' are determined by the nature of the system, i.e. by its proper functions, regardless of its state, akk' is the amplitude of the cor- responding dipole, which would be produced by the proper vibrations ipk^ ^k', if only these two were excited, = and with equal strength lck= Ck' • "v^) The first sum in (19) is of no interest in our investi- gation of the emitted radiation, since it means a component of electric moment that is constant in time. The correctness of our 0i/f-hypothesis has been checked by calculating the akk''^ in those cases where the i/f/e's are sufficiently well defined, namely in the case of the Zeeman and Stark eff"ects. The so-called rules of selection and polarization and the intensity-distribution in these patterns are described by the ^aa's in the following obvious way, and the description is in complete agreement with experiment: 20 WAVE MECHANICS The absence of a line which might be expected to occur (*' selection-rule ") is described by the vanishing of the corresponding akk'y cind of the two other constants relating to the x- and j^-directions. The linear polarization of a line in a definite direction is described by the fact that only the constant akk' relating to this direction differs from zero, whereas the two other constants vanish. In a similar way the circular polarization, say in the x^-plane, is indicated by (1) vanishing of the ^-constant, (2) equality of the x- and ^/-constants, and (3) a phase-difference of 77/2 between the corresponding cosine-functions in equation (19). Finally, the intensity relations between the non-vanish- ing components in the Stark and Zeeman patterns of hydrogen are correctly indicated by the relations between the squares of the ^^/^-'s in question; which is satisfactory, since the assumption that the CkS will be equal for the fine-structure components of one level is very suggestive, notwithstanding our lack of knowledge of the c^s in other respects. Of course it is impossible to set forth in this lecture any of the calculations that led to the results just given; they would fill pages and pages, and are not at all difficult, but very tedious. In spite of their tediousness, it is rather fascinating to see all the well-known but not understood '' rules " come out one after the other as the result of very familiar elementary and absolutely cogent analysis, like e.g. the fact that / cos mcj) cosn^dcf) vanishes unless —_ n m. Once the hypothesis about i/jip has been made, no accessory hypothesis is needed or is possible; none could help us if the " rules " did not come out correctly. But fortunately they do. SECOND LECTURE 21 I think I ought to draw attention to another fact which was only briefly mentioned at the beginning, namely, that the very fundamental " frequency-rule " of Bohr, = — vkk' ""^k— n' {Ek Ek')y 1^ may also be said to be explained by the «/fi/f-hypothesis. Something exists in the atom which actually vibrates with the observed frequency, viz. a certain part of the electric density-distribution or, if you prefer, of l/jl/j. This might lead us to suspect that only the square of its absolute value, and not the ?/f-function itself, has a real meaning. And this suspicion again might arouse the desire to replace the w^ave equation by an equation which describes the behaviour of j/f«/f directly. To remove this desire, I will remind you of a case in which a similar desire might occur for exactly similar reasons; yet all of you will confess that it would be fatal to pursue it. Maxwell's equations describe the behaviour of the electromagnetic vectors. But these are not really accessible to observation. The only things that are observable are the ponderomotive forces, or, if you please, the energy, since the forces are caused by virtual energy- differences. But all these quantities (energy, Maxwellian-stresses) are quadratic functions of the field-vectors. Therefore we might desire to replace Maxwell's equations by others, that determine the observable quadratic functions of the field-vectors directly. But everyone w'ill agree that this would at all events mean an immense complication and that it would not really be possible to do without Max- well's equations. 22 WAVE MECHANICS 6. Derivation of the wave equation (properly speaking) which contains the time. The equation V'^ + ~{E-V)i. = 0, . (13) which we have used for the investigation of the hydrogen atom, only furnishes the distribution in space of the ampHtude of the vibration, the dependence on time always being given bv 2ntEt - i/j e~^ (21) The value of the frequency, E, is present in the equation, so that we are really dealing with a family of equations, each of the members being valid for one particular frequency only. The state of things is exactly the same as in ordinary vibration problems; our equation corresponds to the so-called " amplitude equation " (see section 3, equation (10')), + ~4'-o, ^''l' . . . (10') and not to .... V^-p-^p = 0, (10) from which the former is derived in the manner described above (namely by supposing /) to be a sine-funciion of the time). In our case the problem is to make the analo- gous step in the reverse direction, i.e. to remiove the parameter E from the amplitude equation and introduce time- derivatives instead. This is easily done. Take one of the family (13) (with a particular value of E), then by (21) we have 277/^' ; h) SECOND LECTURE Using this, we get from (13) V20 ^_ ,/,___^ = 0. 23 . (22) The same equation is reached whatever the value of E may have been (for E has been ehminated). Hence equation (22) will be valid for an arbitrary linear aggregate of proper vibrations, i.e. for the most general wavemotion that is a solution of the problem. We may tentatively go a step farther and try to use it also in the case where the potential energy V contains the time-variable explicitly. It is by no means obvious that this is a correct generalization, for terms with F, &c., — might be missing they could not possibly enter into equation (22), in view of the way we have reached this equation. But success will justify our procedure. Of course it would have been nonsense to introduce the V assumption that contained the time explicitly in equation (13), since the condition (21), by which this equation is restricted, would make it impossible to satisfy (13) in the case of an arbitrarily varying F- function. 7. An atom as perturbed by an alternating electric field. This generalization enables us to solve the important problem: how does an atom behave under the influence of an external alternating electric field, i.e. under the influence of an incident wave of light? This is a very important question: for it contains not only the mechanism of secondary radiation and, in particular, of resonanceradiation, but also the theory of the changes of state of the atom under the influence of an incident radiation of appropriate frequency, and in addition the theory of 24 WAVE MFXHANICS refraction and dispersion; for it is well known that — dispersion I mean the phenomenon of a refractive — index is caused by the superposition on the primary radiation of all the secondary wavelets, which every single atom of the body emits under the action of, and in phase with, the primary radiation. If an incident E electric vector causes every atom to emit a secondary wavelet such as would be emitted by a dipole of the electric moment M = aE (23) Z (a being a constant), and if atoms are present in unit volume, then they produce an increase in the refractive index of 27rZa (24) Hence, studying the value of a (which usually depends on the frequency) means studying the phenomena of refraction and dispersion. To investigate the behaviour of an atom in an external V alternating electric field, let us take in equation (22) to be composed of two parts, one describing the internal electrostatic field of the atom, Fq, and one describing the light-field, Aez cos ZTrvt; A, v mean the amplitude and the frequency of the light-field, which we suppose polarized in the direction of z. (The negative sign of the electronic charge has been taken account of; our ^ is a positive number.) Hence equation (22) becomes _ - + = V2^ 1^-''^ ^^'"^ (Fo Aez cos27Tvt)ilj 0. (25) We A shall take to be very small compared with the internal field (described by Vq) and solve the equation A by approximation. If were zero, by assuming (21) we SECOND LECTURE 25 should get back to equation (13) (only with the notation We Vq instead of V). shall assume the problem of the unperturbed atom to be completely solved, its normal- ized proper functions and proper values being ijjk and Ek{=hvk). = Hence the most general solution of (25), when ^4 0, is = .... ilj i:c,rPke'^^^^\ (26) k the CkS being arbitrary complex constants. We A shall try to satisfy equation (25), with also present, by (26), but with the Ck's varying slightly with time (method of variation of constants). Taking this into account, and also the facts that ipk, hvk are proper func- tions and proper values of the unperturbed equation, we easily obtain, by substituting (26) in (25): = ^-^^^ S ^'Aa^'""^' ^^^ 27rvtZckiljke-''''^\ (27) k '*' k This equation will be satisfied if (identically with respect to time) all the coefficients of the expansion of its left-hand side with respect to the complete system of orthogonal functions, ijjk^ are identical with the respective coefficients of the expansion of its right-hand side. Hence multiply by ijji and integrate over the whole space. Put for abbreviation (see section 5): = aki e i \ i ifjk^izdx dy dz. , (20) Then, owing to the normalization and orthogonality of the 0;i's, we get ^ ^ 2 e2-'V,f ci 277Z n ^^^ 2 ^^t aki Ck ^-"'•''t' k (D929) (/= 1,2^3,4,...). . . . (28) 3 26 WAVE MECHANICS This infinite set of ordinary differential equations is equivalent to (27). Isolating q and splitting up the cosine into exponentials, we write it as follows: = + !^ S r; akiCk [^2-^-(.,-.',+ .)^ _|_ ^2 .f (.,-.,-.).] ^ (28') Hitherto we have not made use of any approximation We process. will now do this in two different ways, one leading to the theory of secondary radiation (excluding the case of resonance) and of dispersion, the other furnish- ing the case of resonance and the changes of state of the atom. THIRD LECTURE 8. Theory of secondary radiation and dis- persion. In equation (28') we shall at first assume that all the aggregates which appear in the exponents are large as compared with the order of magnitude of h' This means that the difference between the incident frequency and any one of the frequencies of spontaneous emission is large compared with the frequency that would correspond to the potential energy which the atom acquires in the external field. (Exclusion of exact or near resonance.) With this assumption the equations (28') show that all the time-derivatives of the q's are small compared to the time-derivatives of the exponentials. After having stated this, let us take any one of the ex- ponentials on the right-hand side of any one of the We equations (28'). may assume its coefficient Ck to be constant during a period of the exponential. Hence this term will only cause a small periodic oscillation of the Ci (on the left), which is restored (or nearly so) after 27 28 WAVE MECHANICS the exponential has gone through a period. But the same holds for all the exponentials. Hence all the c's execute a vast number of small oscillations around their mean values, oscillations which would of course vanish We with vanishing A. may therefore replace the t's on the no'/?^-hand side of equation (28') by constants, viz. by their mean values, since by neglecting the small We oscillations here only terms in A^ are dropped. shall write CfP for the said constants. The equations are now We easily integrated. get Hence the /th term in our solution (26) will be: — ^-^i:a,,cA^ + -" . (29) Though we have not yet reached a point that can be compared with experiment, we will give in words the description of what happens, according to equation (29), under the influence of an incident light-wave. Every proper vibration j/f/, whether it is itself excited from the beginning or not, is compelled to execute a multitude of small additional forced oscillations, namely two '' in honour " of every proper vibration ipk that is excited appreciably {ck^ 4= 0). The frequencies of the two forced oscillations that 0/ executes *' in honour " oi ipk, as we i said, are Vk v, i.e. the sum and diff"erence of the incident frequency and the frequency of the " honoured " proper vibration. Their amplitudes are proportional to the amplitudes both of the external field and of the '* honoured " vibration; they also contain as a factor a^h the constant THIRD LECTURE 29 which governs the intensity of the spontaneous emission — of frequency v^ \ yi\. Further, in the two forced ampHtudes two " resonance-denominators " appear, caus- ing one of the two ampHtudes to increase rapidly when the incident frequency approaches the frequency of spontaneous radiation \ Vk— '^^i]- Before forming the complete solution from (26) and (29) we will restrict ourselves to the most important case, viz. that in which only one free vibration is excited, say ipk- Ck^=l cP=0 for l=^k. We may think of i/jk as corresponding to the normal state. Then on the right-hand side of equation (29) the first = term (except for I k) and the summation sign are dropped, and we get for the complete solution (equation (26), in which k is to be replaced by /): = + ^ ^,.^-".' E «,;0, -^ — + -^ . (30) (Note that now the exponentials are independent of the index of summation, /; only two frequencies of forced vibration are present.) To get information about the secondary radiation we form the component * M^ of the resultant electric moment from (30). Neglecting small terms of the second order (proportional to ^^), we find after reduction: = — — M^ e I j iljiljzdxdydz= dkk h (31) I {vi—Vkf—v^ * In general, for an anisotropic atom, there will be an My and M^ an (orthogonal to the polarization of the incident radiation) We as well. will not deal with them here. 30 WAVE MECHANICS The first term (—akk) is independent of the time; it is the constant electric moment due to the excitation of the free vibration i/j^. It is of no interest here. The second term determines the secondary wavelet. It is seen to coincide in frequency with the incident electric force (A cos27Tvt). Its phase is the same or opposite, depend- — ing on whether v '^ vi Vky just as in the classical theory. (This holds if ijjk corresponds to the normal state, — so that vi Vk is always positive; if it is negative, the reverse is true; Kramers' terms of the dispersion formula.) The quantity a of equation (23), which by the expression (24) determines the contribution to the re- fractive index, is found from the second term on the A right-hand side of (31) by dropping cos27Tvt. The — — denominators (v/ VkY ^^ furnish the phenomenon of anomalous dispersion in the neighbourhood of all those emission (or absorption) frequencies that involve the — index k of ipk remember that we supposed only this one free vibration to be excited. The quantity aki^ in the numerator is the same as that which determines the — intensity of spontaneous emission i^^ i^z . In all these | | respects the formula is a complete copy of the old Helmholtz formula (supplemented by Kramers' " negative " terms) and is thought to be in complete agreement with experiment. Two additional points are worth mentioning. You know that Thomas and Kuhn formed a hypothesis con- cerning the sum of all the coefficients in the dispersion formula, in our case 2 According to them it is to be equal to the value of the THIRD LECTURE 31 coefficient for one elastically bound electron, i.e. it must be equal to 4:7T^m (multiplied by one, in our case, for we are dealing with the o;z£'-electron atom; in general, multiplied by an integer). The equality of the two above-mentioned — quantities can be proved for our dispersion formula but the proof is a little lengthy, and I will therefore omit it. The second remark is the following. Perhaps you remem.ber the statement, first made by Smekal, that there should also exist secondary radiations, whose frequencies differ from the frequency v of the incident radiation (therefore without phase relation, therefore without influence on the refraction phenomenon). The frequencies expected are Secondary radiations of precisely these frequencies are furnished by the present theory, if we give up our sim- plifying assumption that only one free vibration is excited, and suppose at least two of them, say ipk and to i/r/j-, be present. 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. At the beginning of the last section we had to make the assumption that all the aggregates like are of appreciable size, which means that the frequency 32 WAVE MECHANICS of the incident light, v, is excluded from the immediate neighbourhood of any natural frequency of the atom We under consideration. will now consider an incident frequency which is very close to one of the natural frequencies. To fix our ideas, let — + > ^yfe J^z i^ be very small and vi Vk (" very small " means: of the order of magnitude of Aukijh or smaller, possibly vanishing). Returning to equation (28'), you will now find on the right-hand side of this system of equations altogether two exponentials which vary slowly, viz. the former appearing in the /th equation, the latter in the ^th equation. These terms (as we shall see presently) now cause very appreciable " secular " changes in the two A quantities c^ and q, however small the amplitude of the incident wave may be. All the other exponentials will only cause small periodic disturbances, as before. It is therefore reasonable to drop them altogether, since we are now dealing with a much coarser phenomenon (viz. We appreciable secular variations of c^ and q). might even suppose all the other c's to be zero; this would have no effect, since they are certainly constant within the degree of accuracy we are aiming at. For determining Ck and Ci we get from (28') the two simple equations Ck=i(JCie ''\ with the abbreviations T^Aaki , (32) /.JON THIRD LECTURE 33 To solve them, we introduce new variables x, y by putting ci=xe-, = Ck ye '^, . . (34) The result can be written i€\ lay. (d _u-\\y = i(jx. \dt 2/ These equations have constant coefficients and are readily solved by familiar methods. The solution can be written in the following form: (35) with the abbreviations Y+ = vJ + ''^ f^=^' (^6) whereas p, p\ (/>, ^' are arbitrary real constants, non- We negative if you like. can put (35) in the form: x=e 2 = y e 2 [{p-irp.p')co^e-\-i{p-iip')s>me]A — + + [(/xp p') cDS^ z(/xp p') sm^],J with the abbreviation e=y,+i+i:. . . . (38) From (37) we can easily form the squares of the absolute values of x and y, that is (by equation (34) ) of ci and Ck, and we can thus get information about the varying dis- 34 WAVE MECHANICS tribution of intensity between the two vibrations in — We question which is the point of main interest. obtain = h = - + U/ 1' I' (P H^Py ^f^Pp' cos2 9, Ck\'=\y\'={^p-py-^4fjipp'sm^e. ' ^^^^ The sum of the intensities is constant^ as might have been anticipated. It may be taken to consist of three parts, two " portions " fixed invariably to the two vibration- levels, the third (viz. 4/x/)/)') oscillating slowly between them. To ^-k. our ideas, let us take the case where at a certain time all the intensity was stored up in one vibration, say the lower one, Ck. Choosing the corre- = sponding value of t so as to make cos ^ 0, this requires = ^. P' We then find for the ratio between the oscillating portion of the intensity and its total amount (by using the fact, obvious from (36), that We = see that when € the total intensity is oscillating. = By (33), e means the case of sharp resonance. If the resonance is not complete, then (40) shows that only a certain fraction of the intensity oscillates, and that this fraction becomes inappreciable when the lack of re- sonance, e, becomes large compared with the quantity THIRD LECTURE 35 a defined by (33). (The order of magnitude of o- is the potential energy (divided by h) which the atom acquires in the electric field of the light- wave, owing to the electric moment which is due to the co-operation of the Ath and /th modes of vibration.) The quantity o- would, in a certain sense, give a measure of the natural sharpness of the resonance-line, if it were possible to form a universal A We idea of the amplitude of the incident light. shall not enter upon this question here. The theory put forward here in its rough features de- scribes both the change of state of the atom produced by radiation of appropriate frequency and the appearance of resonance-radiation. For of course the presence of the two vibrations ijjk and ipi will give rise to their natural emission. It is worth while mentioning that on account of the exponentials appearing in equation (34) this — emission should not have exactly the frequency vi Vky but a frequency exactly equal to v, the frequency of the incident light- wave. 10. Extension of wave mechanics to systems other than a single mass -point. Hitherto we have applied the method of wave me- chanics only to a very simple system, viz. a single mass- point moving in a field of force which was either constant We or varying with the time. will now proceed to a quite We arbitrary mechanical system. might have done this before; all that has been said about the influence of an alternating field would apply with very slight modifi- cation to an arbitrary system, e.g. to the many-electron atom. But I thought it better to have a clear and simple case before our mental eye. The derivation of the fundamental wave equation 36 WAVE MECHANICS put forward in the first lecture is very easily generalized to a quite arbitrary system, the only difference being that the *' space " in which the wave-propagation takes place is no longer ordinary three-dimensional space but the ** configuration space ". Let us recall the Hamilton-Maupertuis principle from which we started, namely, SJ'2Tdt = .... (1) and which we transformed into Sr^2m{E-V)ds = 0, ... (2) JA by putting 2r- mw' = m (jY = 2{E -V) = j^ V2mXE^T). We then compared it with Fermat's principle for a wave- propagation: which led us to aff^o, ..... (3) J^ u ^ V2m{E-V) ... (4) Now, in general T is not of the simple form ^ (^) but 2T=I.^b„q,q„ Ik . . . (41) where the hi^s, are functions of the generalized co- We ordinates qi. now define a line-element ds in the generalized ^-space by or = ds'- i:i.bikdqidqk. . . . (42) THIRD LECTURE 37 The generalized non-Euclidean geometry, which is de- fined by the latter formula, is exactly the one which Heinrich Hertz used in his famous mechanics and which allowed him to treat the motion of an arbitrary system formally as the motion of a single mass-point (in a non- Euclidean, many-dimensional space). Introducing this geometry here, we easily see that all the considerations of the first lecture which led us to the fundamental wave equation may be transferred, even with a slight formal w = simplification, viz. that we have to put 1. In exactly the same way as before we obtain E V2{E- V) and finally for the wave (or rather amplitude) equation: V^ + ^{E-V)^=0. . . (43) For the wave equation properly speaking we get, just as before (section 6), „„ , 4:771 ; Stt^V , ^ ,i .X But, of course, V^ is now to be understood not as the simple Laplacian in three dimensions nor as the simple Laplacian in a many-dimensional Euclidean space (i.e. the sum of the second derivatives with respect to the single co-ordinates), but it is to be understood as the well-known generalization of the Laplacian in the case of a general Hne-element like (42). In the treatment of general problems we can usually avoid writing down the explicit expression for this operation; we need only know that it is a self-adjoint differential operator of the second order. (Never mind whether you know what '' self- 38 WAVE MECHANICS adjoint " means, it is of no importance for the moment.) Yet for the sake of completeness I will put down the general expression for V^. Let aik be the minor corres- sponding to hiky divided by the determinant S zb ^/y^- Let « be the determinant of the aikS. Then ^''^ ^'-'^'^kk'^'i^'^)- • In the case of a single mass-point of mass 7;z, treated —m in Cartesian co-ordinates, this reduces to times the + + elementary V^-operator (viz. d^/dx^ d^/dy^ d^/dz^). Or, if you chose to describe the motion of a single masspoint by any other co-ordinates, e.g. polar or elliptic, you — would get times the expression for the elementary V^ transformed to those co-ordinates. If the system consists of n free mass-points, you get the sum of their elementary V^-operators each divided by the appropriate mass. The theory in its present form is applicable to systems of any number of degrees of freedom more than, equal to, or less than, three. I shall give a rapid account of a few examples without going through the details of calculation unless they present some physical interest. 11. Examples: the oscillator, the rotator. Take the ow^- dimensional harmonic oscillator. The expression for the energy in ordinary mechanics may be taken to be (we have expressed the coefficient of the potential energy in terms of the classical proper frequency Vq which it THIRD LECTURE 39 produces). This easily leads to the amplitude equation: It can be shown that this equation has solutions which are finite along the real ^-axis, for the following values of E only: = + = E„ {n i)hvo; 72 0,1,2,3. . (46) The proper functions are the so-called Hermite ortho- gonal functions = i/r„ (2"«!)"2^"2/f„(A;) . . r47^ with •^W'? A H„{x) is the so-called wth Hermite polynomial. graph of the first five functions (47) is given in the figure. >+CG The first five proper vibrations of the Planck oscillator according to undu- — + latory mechanics. Outside of the region 3 < x_< 3 represented here, all five functions approach the x-axis in monotonia fashion. Though theoretically they extend to infinity, they are practically restricted by the exponential to a domain of the order of magnitude of the amplitude of the corresponding 40 WAVE MECHANICS We classical mass-point. (This is very easy to prove.) have not discussed the physical meaning of our generalized i/f-function. Yet the following statement is of interest. If the i/r„'s were the proper functions of a one-electron problem and q one of the rectangular co-ordinates, we would (following our j/fj/f-hypothesis) estimate the in- — tensity of emission of frequency -j\En Ek\y polarized in the direction of ^, by the square of the integral j qi/jki/j^^dq. If we try to do the same here^ we get a most satisfactory result, viz. the integral vanishes^ unless \k-n\ = l. This means that all the emission frequencies except We 1 . vq are excluded. shall return later to the question of the physical meaning of in the general case. Take as a second example another one-dimensional problem: the simple rotator with its axis fixed in space. Here all the energy is kinetic, viz.: 2 \dt A = = where moment of inertia, ^ angle of rotation. The amplitude equation becomes which has the solutions: sin ^ COS Obviously ifj must be restricted to be periodic in ^ with ) THIRD LECTURE 41 period 27r. Hence the coefficient of (/> must be an integer; this condition furnishes the proper values ^" = £fi' « = 0.1.2,3 , (48) in complete agreement with the older form of quantum theory. Let us try to get an estimate for the intensity of radiation in the same formal way as before. If, in ordinary mechanics, an electrified particle were fixed to the rotator at a distance a from the centre of gravity, its rectangular co-ordinates would be x\ /cos\ , Now form Since the p^roduct of the first two ^^ \ cosj functions can . ^ always be expressed by the sum or difference of \ + {n k)(f>, it is easily recognized that none of the eight quantities comprised in the above formula diflfers from + — zero, unless either ;z ^ or \n k\ is unity; or, | | what amounts essentially to the same, unless \n-k\ = l. This is the well-known selection-rule for the rotator. It is interesting to treat the rotator again without the We assumption that its axis is rigidly fixed in direction. find for the amplitude equation Here V\,/, means that part of the elementary V--operator D ( 9:d9 4 42 WAVE MECHANICS (when expressed in polar co-ordinates) that contains the differentiations with respect to the angles 9, only. It is known that the above equation only has finite singlevalued solutions w^hen the constant is the product of two successive integers: = —^2" =^{^i+ 1); w 0, 1,2, . . ., and that the solution is a spherical harmonic of order n, + (The proper value £„ is {2n l)-fold degenerate, since + there are 2w 1 independent spherical harmonics of order n.) This furnishes the proper values this means essentially that *' half-integers " are to be inserted in place of n in the " classical " formula (48). + = + — (For «(w 1) (w J)^ i, and a common constant in all the E„'s cancels out in forming their differences.) It is known that the representation of band-spectra very often compelled the use of " half-integers ", and it seems that all of them are compatible with the new formula. (Of course formula (49) is the correct one to use, and not (48), because the axis of a molecule is never rigidly fixed.) The selection rule comes out in exactly the same way as in the former case, only by a more troublesome calcu- lation. FOURTH LECTURE 12. Correction for motion of the nucleus in the hydrogen atom. In the first lecture we treated the hydrogen atom as a one-body problem, as if the nucleus were fixed in space. Ill ordinary mechanics it is well known that if we start with the problem of two bodies (of masses m and M), we can split it in two, viz.: (1) Uniform rectilinear motion of the centre of gravity (inertial motion). (2) Keplerian motion around a fixed centre of a body, with the " combined mass " /^, such that l-i + m (50) According to Bohr's theory, this refined treatment of the hydrogen atom is quantitatively supported by the slight difference in frequency between the Helium + -lines and those hydrogen-lines which would exactly coincide with them if the nucleus had infinite mass. (In other words, the slight difference between the Rydberg constant for He + H and for is quantitatively accounted for by taking into account the slight movement of the nucleus; Sommerfeld.) We meet with exactly the same state of affairs in wave mechanics. The six-dimensional amplitude equa- tion for the two-body problem is: ^V,2^ + l^^i^-V ^-fiE-V)^=0. (51) 43. 44 WAVE MECHANICS By Vi^ and Vg^ we mean the elementary Laplacians with respect to the co-ordinates of the electron {x^, j^, z^) and V of the nucleus {x2, y2, ^2)- About we need only make the assumption that it depends on only. Now, instead of x^, . . ., z^, introduce the co- ordinates of the centre of gravity (f , 7^, f) and the relative m M We co-ordinates of with respect to (say Xy j, z). can easily prove that The meaning of the V-'s is obvious; /x is given by (50). By inserting this in (51) we get an equation which can be split up by supposing ijj to be the product of a function of f , 7?, ^ only (say ^) and one of x, y, z only (say x). In the splitting up an arbitrary constant is introduced, which is represented by Et in the following equations. For cf) we get ;;^V^.„.^ + '#'^ = 0, . (62) and for x j^V\,,^X + ^(E-E,-V)x = 0. (53) The former describes the motion of the centre of gravity under no forces, according to wave mechanics; the constant Et corresponds to its translational energy and E — can have any non-negative value. Et corresponds to the internal energy. The second equation is exactly that of the one-body problem for a mass-point with mass /x moving in a fixed field V. Hence for the proper values corresponding to the internal energy there will be no FOURTH LECTURE 45 m difference other than that is replaced by /x (see (14')) in the formula for the Rydberg constant. Thus Sommerfeld's important result, mentioned above, is re-stated in wave mechanics. Owing to the analytical simplicity of this deduction, there has not been much ado about it in the literature. But it really is one of the most immediate proofs that there must be something true in the many- — dimensional wave-treatment however irritating the latter may be at first. 13. Perturbation of an arbitrary system. The theory of the perturbation of an arbitrary system really presents no new features as compared with the perturbation theory of the one-electron atom, a special case of which has been discussed in sections 7-9; but we shall widen our outlook by stating it afresh in a concise form. The general wave equation (44) of section 10 can be written: We H will write for the operator OTT V) (Fas an operator means: " to multiply by Then by (43), section 10, the proper functions ifjk are precisely those which are reproduced by the operator H, apart from a multiplying constant, which is the proper value: ffM = £,^, (55) Equation (54) takes the simple form 4>~Hm (56) : 46 WAVE MECHANICS Now, adding to F a small perturbing field, which may or may not contain the time explicitly, means altering the H H operator slightly. (Of course an alteration of might also be produced in another way, e.g. by altering one of the masses, &c. It will do no harm if this more general We case is included in our treatment.) shall call the H H\ altered operator -\- bearing in mind that H' is to We be a '' small " operator. have to solve i.= ^-p{H[>l.]+H'W). . . (57) Tentatively substituting = S<:,^,e'^; ^.= 5, • • (58) with slowly varying time-functions Ck^ we obtain in the first instance k nk This equation will be satisfied if it is orthogonal to all the ipis *. Multiply by j/j/ and integrate over the whole configuration-space ci=-j^Y.Ckaike /= , 1, 2, 3, 4, . . ., (59) where = aik j (lqH'[iPk]iph • • • (60) and / dq always means a multiple integral over the whole configuration-space. The af,i's are s?nall quantities. We will suppose the perturbation to be conservative. We * take it for granted that with respect to the completeness and orthogonaUty of the proper functions the general case behaves We like the simple hydrogen case. That is quite safe. also, as there, avoid encumbering our formulae by exphcitly taking account of a continuous spectrum of proper values. FOURTH LECTURE 47 Then the aki's are constants; just as in the special cases treated before, only the exponentials with vanishing exponent will cause appreciable variations of the q's. First take the system to be non - degenerate. Then, dropping the other terms, which only furnish slight oscillations, you get, for every q, c,= ^cr, c,= c?e~\ . . (61) which, if you substitute it in (58), merely means that the frequency is slightly altered by the amount h' Now take a case of degeneracy. Let the amplitudes Ci, Ci+i, . . . , Q+a-i belong to a different proper functions, all belonging to the same proper value Ei, or proper frequency v/. Then in each of the equations relating to them you will have not only one, but a, vanishing exponents, which give rise to secular changes. Hence these a amplitudes wdll be determined by the following set of equations: = = — L ci+p -r- S Ci+^ai+pj+^; /) a 0, 1, 2, . . ., (62) These equations show that under the influence of a slight perturbation there will in general be an exchange of amplitudes between degenerate modes of vibration which belong to the same proper value. It is correct to talk of an exchange, since it is easily proved from equation (62) that = '2 |r/+p|' const. p= Yet when thinking of this exchange we must remember 48 WAVE MECHANICS = — that the set of proper functions ipupip a 0, 1, . . . , 1) is arbitrary up to an orthogonal hnear substitution of determinant 1. This induces a similar substitution of the amplitudes q. Given a definite perturbation, i.e. definite values of the quantities fl/+A,/+p, it is always possible to find at least one orthogonal substitution of the j/f/+p's which brings the equations (62) into the simple form (61) of the non- degenerate case. Then these particular proper functions, selected in a way that suits this particular form of perturbation, will under its influence have constant amplitude-squares, but will in general belong to slightly different proper frequencies. The a-fold proper value has been split up into a slightly differing proper values; the degeneracy is removed by the disturbing field, and the particularly chosen proper functions of the degenerate problem are the non- degenerate proper functions " in zero approximation " to the single proper values of the perturbed problem. The a slight alterations in proper value can be shown to be the a roots of the " '' secular equation — Ull X, (llji-l, ' ' ', «/, /+aJ = Of course it may happen that these roots are not all We different; a certain degeneracy is then retained. may either say that the members of an arbitrarily chosen set of the degenerate functions all vibrate with the un- — perturbed frequency, but exchange their amplitudes or that the members of the appropriately chosen set have constant amplitudes, but that each function has a slightly different frequency: these two assertions are of course FOURTH LECTURE 49 — — identical. For as we may put it either: a vibration of varying amplitude has not really got the frequency which we ascribe to it; or: two or more slightly different frequencies, when superimposed, lead to a " beat phenomenon ", i.e. to a varying amplitude. 14. Interaction betw^een two arbitrary systems. Take now two arbitrary systems, at first without interaction, one of which is described according to wave mechanics (see equation (56) ) by and the other by Multiply the first by ^, the second by ip, and add the resulting equations; you get H since the operator L does not affect cf) and does not affect ip. The latter equation is the wave equation of the " combined system ", i.e. of the system formed by mentally uniting the two systems to form one. (The process is exactly the reverse of what is so often done in " splitting up " an equation by supposing the solution to be the product of two functions, dependent on different individual variables.) The proper functions of the com- bined system are the products of any one of the proper functions of the first system and any one of the second system. The proper value that belongs to such a product is easily seen to be the sum of the respective proper values. (This corresponds to the additivity of energy in ordinary 50 WAVE MECHANICS mechanics.) By the addition of proper values a 7iezo degeneracy may be caused in the combined system, even though the single systems were non-degenerate. (Let us suppose the latter case, for the sake of simplicity.) Let £", E' be two proper values of the first system, F, F' two of the second system, and suppose that £' + F = F + i^=G or E-E' = F-r. Hence: if a common difference of proper values exists between the two systems, it will give rise to a two-fold G degenerate proper value of the combined system. For simplicity's sake, suppose that other relations of the same kind are absent, and now suppose that a slight interaction of the two systems takes place, changing the H H ^ L operator -\- into L -\- T, where T will of course F'- contain the variables both of the first and of the second system. Then the amplitudes belonging to E -^ F' and F to E' -}- will show a slow secular interchange, all the others remaining essentially constant. The sum of the squares of the two amplitudes in question is also constant. Interpreted in the single systems, this cannot very well have any other meaning but that e.g. the amplitude of F increases at the expense of that of F' and, so to speak, to compensate for the amplitude of E' increasing at the expense of that of E. This seems to be the appropriate wave-mechanical description of what in the older FOURTH LECTURE 51 form of the quantum theory was called the transfer of a — — quantum of energy E E' {= F F') from one system to the other. 15. The physical meaning of the generalized i/f-function. Perhaps the latter conclusions are obscured by the fact that we have hitherto avoided putting forward any definite assumption as to the physical interpretation of the func- tion ip{q^, q^y qn, . . . , t) relating to a system whose configuration in terms of ordinary mechanics is described by the generalized co-ordinates ^1, ^2? • • • > ?«• This interpretation is a very delicate question. As an obvious generaliza- tion of the procedure of spreading out the electronic charge according to a relative density function «/f j/f (which furnished satisfactory results in the one-electron problem; see section 5), the following view would present itself in the case of a general mechanical system: the real natural system does not behave like the picture which ordinary mechanics forms of it (e.g. a system of point- charges in a definite configuration), but rather behaves like what would be the result of spreading out the system, described by ^1, . . . , 9„, throughout its con- figuration-space in accordance with a relative density function 0j/f. This would mean that, if the ordinary mechanical picture is to be made use of at all, the actual system behaves like the ordinary mechanical picture, present in all its possible configurations at the same time, though '* stronger " in some of them than in others. I maintained this view for some time. The fact that it proves very useful can be seen from the one-electron problem (see section 5). No other interpretation of the i/f-function is capable of making us understand the large — 52 WAVE MECHANICS amount of information which the constants aki furnish about the intensity and polarization of the radiation. Yet this way of putting the matter is surely not quite satisfactory. For what does the expression '* to behave like " mean in the preceding sentences.^ The '' behaviour " of the j/f-function, i.e. its development in time, is governed by nothing like the laws of classical mechanics; it is governed by the wave-equation. An obvious statistical interpretation of the ^-function has been put forward, viz. that it does not relate to a single system at all but to an assemblage of systems, ijj determining the fraction of the systems which happen to be in a definite configuration. This view is a little unsatisfactory, since it oflFers no explanation whatever why the quantities aki yield all the information which they do yield. In connexion with the statistical interpretation it has been said that to any physical quantity which would have a definite physical meaning and be in principle {principiell) measurable according to the classical picture of the atom, there belong definite proper values (just as e.g. the proper values Ek belong to the energy); and it has been said that the result of measuring such a quantity will always be one or the other of these proper values, but never anything intermediate. It seems to me that this statement contains a rather vague conception, namely that of measuring a quantity (e.g. energy or moment of momentum), which relates to the classical picture of the atom, i.e. to an obviously wrong one. Is it not rather bold to interpret measurements according to a picture which we know to be wrong? May they not have quite another meaning according to the picture which will finally be forced upon our mind? For example: let a beam of electronic rays pass through a layer of mercury vapour, and FOURTH LECTURE 53 measure the deflection of the beam in an electric and in a magnetic field before and after the beam has traversed the vapour. According to the older conceptions this is interpreted as a measurement of diflferences of energy- levels in the mercury atom. The wave-picture furnishes another interpretation, namely, that the frequency of part of the electronic waves has been diminished by an amount equal to the difference of two proper frequencies of the mercury. Is it quite certain that these two interpretations do not interfere with one another, and that the old one can be maintained together with the new one? Is it quite certain that the conception of energy, indispensable as it is in macroscopic phenomena, has any other meaning in micro-mechanical phenomena than the number of vibrations in h seconds? QA 927.S33 3 9358 00183735 7 QA 927.S33 3 9358 00183735 7 %