(20)
then
(21)
If we put, in analogy with (19),
(22)
then a′kn′ = − Fakn , and by carrying out the proposed integration we find,
(23)
for the resulting electric moment, to which the secondary radiation, caused by the incident wave (20), is to be attributed.
The radiation depends of course only upon the second (time-variable) part, while the first part represents the time-constant dipole moment, which is possibly connected with the originally existing free vibration. This variable part seems fairly promising and may meet all the demands we are accustomed to make on a “dispersion formula”. Above all, let us note the appearance of those so-called “negative” terms, which—in the usual phraseology—correspond to the probability of transition to a lower level (En < Ek), and to which Kramersfu was the first to direct attention, from a correspondence standpoint. Generally, our formula—despite very different ways of thought and expression—may be characterised as really identical in form with Kramer’s formula for secondary radiation. The important connection between akn , bkn , the coefficients of the secondary and of the spontaneous radiation, is brought out, and indeed the secondary radiation is also described accurately with respect to its condition of polarisation.fv
I would like to believe that the absolute value of the scattered radiation or of the induced dipole moment is also given correctly by formula (23), although it is obviously within the bounds of possibility that an error in the numerical factor may have occurred in applying the heuristic hypothesis introduced above. At any rate the physical dimensions are right, for from (18), (19), (21), and (22) akn and bkn are electric moments, since the squared integrals of the proper functions were normalised to unity. If ν is far removed from the emission frequency in question, the ratio of the induced to the spontaneous dipole moment is of the same order of magnitude as the ratio of the additional potential energy Fakn to the “energy step” Ek − En .
§ 3. SUPPLEMENTS TO § 2. EXCITED ATOMS, DEGENERATE SYSTEMS, CONTINUOUS SPECTRUM
For the sake of clearness, we have made some special assumptions, and put many questions aside, in the preceding paragraph. These have now to be discussed by way of supplement.
First: what happens when the light wave meets the atom, when the latter is in a state in which not merely one free vibration, uk, is excited as hitherto assumed, but several, say two, uk and ul ? As remarked above, we have in the perturbed case simply to combine additively the two perturbed solutions (16) corresponding to the suffix k and the suffix l , after we have provided them with constant (possibly complex) coefficients, which correspond to the strength presumed for the free vibrations, and to the phase relationship of their stimulation. Without actually performing the calculation, we see that in the expression for ψψ and also in the expression (23) for the resulting electric moment, there then occurs not merely the corresponding linear aggregate of the terms previously obtained, i.e. of the expressions (17) or (23) written with k, and then with l. We have in addition “combination terms”, namely, considering first the greatest order of magnitude, a term in
(24)
which gives again the spontaneous radiation, bound up with the coexistence of the two free vibrations; and secondly perturbing terms of the first order, which are proportional to the perturbing field amplitude, and which correspond to the interaction of the forced vibrations belonging to uk with the free vibration ul—and of the forced vibrations belonging to ul with uk. The frequency of these new terms appearing in (17) or (23) is not ν but
(25)
as can easily be seen, still without carrying out the calculation. (New “resonance denominators”, however, do not occur in these terms.) Thus we have to do here with a secondary radiation, whose frequency neither coincides with the exciting light-frequency nor with a spontaneous frequency of the system, but is a combination frequency of both.
The existence of this remarkable kind of secondary radiation was first postulated by Kramers and Heisenberg (loc. cit.), from correspondence considerations, and then by Born, Heisenberg, and Jordan from consideration of Heisenberg’s quantum mechanics.fw As far as I know, it has not yet been demonstrated experimentally. The present theory also shows distinctly that the occurrence of this scattered radiation is dependent on special conditions, which demand researches expressly arranged for the purpose. Firstly, two proper vibrations uk and ul must be strongly excited, so that all experiments made on atoms in their normal state—as happens in the vast majority of cases—are to be rejected. Secondly, at least one third state of proper vibration must exist (i.e. must be possible—it need not be excited), which leads to powerful spontaneous emission, when combined with uk as well as with ul. For the extraordinary scattered radiation, which is to be discovered, is proportional to the product of the spontaneous emission coefficients in question (aknbln and alnbkn). The combination (uk, ul ) need not, in itself, cause a strong emission. It would not matter if—to use the language of the older theory—this was a “forbidden transition”. Yet in practice we must also demand that the line (uk, ul) should actually be emitted strongly during the experiment, for this is the only means of assuring ourselves that both proper vibrations are strongly excited in the same individual atoms and in a sufficiently great number of them. If we reflect now that in the powerful term-series mostly examined, i.e. in the ordinary s-, p-, d-, f-series, the relations are generally such that two terms, which combine strongly with a third, do not do so with one another, then a special choice of the object and conditions of the research seems really necessary, if we are to expect the desired scattered radiation with any certainty, especially as its frequency is not that of the exciting light and thus it does not produce dispersion or rotation of the plane of polarisation, but can only be observed as light scattered on all sides.
As far as I see, the above-mentioned dispersion theory of Heisenberg, Born, and Jordan does not allow of such reflections as we have just made, in spite of its great formal similarity to the present one. For it only considers one way in which the atom reacts to incident radiation. It conceives the atom as a timeless entity, and up till now is not able to express in its language the undoubted fact that the atom can be in different states at different times, and thus, as has been proved, reacts in different ways to incident radiation.fx
Let us turn now to another question. In §2 the collective proper values were postulated to be discrete and different from one another. We now drop the second hypothesis and ask: what is altered when multiple proper values occur, i.e. when degeneracy is present? Perhaps we expect that complications then arise, similar to those we met in the case of a time-constant perturbation (Part III. § 2), i.e. that a system of proper functions of the unperturbed atom, suited to the particular perturbation, must be defined by the solution of a “secular equation”, and applied to carry out the perturbation calculation. This is indeed so in the case of an arbitrary perturbation, represented by r (x , t) as in equation (5), but not so in the case of a perturbation by a light wave (equation (6))—at any rate, for our usual first approximation, and as long as we suppose that the light frequency ν does not coincide with any of the spontaneous emission frequencies considered. Then the parameter value in the double equation (13), for the amplitudes of the perturbed vibrations, is not a proper value, and the pair of equations has always the unambiguous pair of solutions (14), in which no vanishing denominators occur even when Ek is a multiple value. Thus the terms in the sum for which En = Ek are not, as might be thought, to be omitted, any more than the term for n = k itself. It is worth noticing that through these terms—if one of them occurs really, i. e. with non-vanishing akn—the frequency ν = 0 also appears among the resonance frequencies. These terms do not, of course, contribute to the “ordinary” scattered radiation, as we see from (23), since Ek − En = 0.
The simplification, that we do not require to consider specially any possible degeneracy present, at least in a first approx
imation, is always availablefy when the time-averaged value of the perturbation function vanishes, or what is the same thing, when the latter’s Fourier expansion in terms of the time contains no constant, i.e. time-independent, term. This is the case for a light wave.
While our first postulation about the proper values—that they should be simple—has thus shown itself to be really a superfluous precaution, a dropping of the second—that they should be absolutely discrete—while leading to no alterations in principle, brings about, however, very considerable alterations in the external appearance of the calculation, inasmuch as integrals taken over the continuous spectrum of equation (1′) are to be added to the discrete sums in (14), (16), (17), and (23). The theory of such representations by integrals has been developed by H. Weyl,fz and though only for ordinary differential equations, the extension to partials is permissible. In all brevity, the state of the case is this.ga If the homogeneous equation belonging to the non-homogeneous equations (13), i.e. the vibration equation (1ʹ) of the unperturbed system, possesses in addition to a point-spectrum a continuous one, which stretches, say, from E = a to E = b, then an arbitrary function f (x) naturally cannot be developed thus,
(26)
in terms of the normalised discrete proper functions un (x ) alone, but there must be added an integral expansion in terms of the proper solutions u(x, E), which belong to the proper values a ≤ E ≤ b , and so we have
(27)
where to emphasize the analogy we have intentionally chosen the same letter for the “coefficient function” φ(E) as for the discrete coefficients φn. If now we have normalised, once for all, the proper solution u(x, E ) by associating with it a suitable function of E , in such a way that
(28)
according to whether E belongs to the interval E′, E′ + Δ or not, then in (27) under the integral sign we substitute from
(29)
wherein the first integral sign refers as always to the domain of the group of variables x.gb Assuming (28) to be fulfilled and expansion (27) to exist—which statements are proved by Weyl for ordinary differential equations—the definition of the “coefficient functions” from (29) is almost as obvious as the well-known definition of the Fourier coefficients.
The most important and difficult task in any concrete case is the carrying out of the normalisation of u(x, E), i.e. the finding of that function of E by which we have to multiply the (as yet not normalised) proper solution of the continuous spectrum, in order that condition (28) may be satisfied. The above-quoted works of Herr Weyl contain very valuable guidance for this practical task, and also some worked-out examples. An example from atomic dynamics on the intensities of band spectra is worked out by Herr Fues in a paper appearing in the present issue of Annalen der Physik.
Let us apply this to our problem, i.e. to the solution of the pair of equations (13) for the amplitudes. w± of the perturbed vibrations, where we postulate as usual that the one excited free vibration, uk, belongs to the discrete point-spectrum. We develop the right-hand side of (13) according to the scheme (27) thus,
(30)
in which is given by (15), and from (29) by
(15′)
If we imagine expansion (30) put into (13), and then expand also the desired solution w ±(x) similarly in terms of the proper solutions un(x) and u(x, E), and notice that for the last-named functions the left side of (13) takes the value
or
then by “comparison of coefficients” we obtain as the generalisation of (14)
(14′)
The further procedure is completely analogous to that of § 2.
Finally, we get as additional term for (23)
(23′)
Here, perhaps, we may not always change the order of integration without further examination, because the integral with respect to ξ may possibly not converge. However, we can—as an intuitive makeshift for a strict passage to the limit, which may be dispensed with here—decompose the integral ∫ba into many small parts, each having a range Δ, which is sufficiently small to allow us to regard all the functions of E in question as constant in each part, with the exception of u(x , E ), for we know from the general theory that its integral cannot be obtained through such a fixed partition, which is independent of ξ . We can then take the remaining functions out of the partial integrals, and as additional term for the dipole moment (23) of the secondary radiation, obtain finally exactly the following,
(23”)
where
(22ʹ)
(19′)
(please note the complete analogy with the formulae with the same numbers but without the dashes in §2).
The preceding sketch of the calculation is of course only a general outline, given merely to show that the much-discussed influence of the continuous spectrum on dispersion, which experimentgc appears to indicate as existing, is required by the present theory exactly in the form expected, and to outline the way in which the calculation of the problem is to be tackled.
§ 4. DISCUSSION OF THE RESONANCE CASE
Up till now we have always assumed that the frequency ν of the light wave does not agree with any of the emission frequencies that have to be considered. We now assume that, say,
(31)
and we revert, moreover, to the limiting conditions of §2 for the sake of simplicity (simple, discrete proper values, one single free vibration uk excited). In the pair of equations (13), the proper value parameter then takes the values
(32)
i.e. for the upper sign there appears a proper value, namely, En. The two cases are possible. Firstly, the right side of equation (13) multiplied by ρ(x), may be orthogonal to the proper function un(x ) corresponding to En, i.e. we have
(33)
which means, physically, that if uk and un exist together as free vibrations they will give rise to no spontaneous emission or to one which is polarised perpendicularly to the direction of polarisation of the incident light. In this case the critical equation (13) also again possesses a solution, which now, as before, is given by (14), in which the catastrophic term vanishes. This means physically—in the old phraseology—that a “forbidden transition” cannot be stimulated through resonance, or that a “transition”, even if not forbidden, cannot be caused by light which is vibrating perpendicularly to the direction of polarisation of that light which would be emitted by the “spontaneous transition”.
Otherwise, secondly, (33) is not fulfilled. Then the critical equation possesses no solution. Statement (10), which assumes a vibration which differs very little—by quantities of the order of the light amplitude F—from the originally existing free vibration, and is the most general possible under this assumption, thus does not then lead to the goal. No solution, therefore, exists which only differs by quantities of the order of F from the original free vibration. The incident light has thus a varying influence on the state of the system, which bears no relation to the magnitude of the light amplitude. What influence? We can judge this, still without further calculation, if we start out from the case where the resonance condition (31) is not exactly but only approximately fulfilled. Then we see from (16) that un (x) is excited in unusually strong forced vibrations, on account of the small denominator, and that—not less important—the frequency of these forced vibrations approaches the natural proper frequency En/h of the proper vibration un. (All this is, indeed, very similar to, yet in a way of its own different from, the resonance phenomena encountered elsewhere; otherwise I would not discuss it so minutely.)
In a gradual approach to the critical frequency, the proper vibration un , formerly not excited, whose possible existence is responsible for the crisis, is stimulated to a stronger and stronger degree, and with a frequency more and more closely approaching its own proper frequency. In contradistinction to ordinary resonance phenomena there comes a point, and that even before the critical frequency is reached, where our solution does not represent the circumstances correctly any longer, even under the assumption that our obviously “un
damped” wave postulation is strictly correct. For we have in fact regarded the forced vibration w as small compared with the existing free vibration and neglected a squared term (in equation (11)).
I believe that the present discussion has already shown, with sufficient clearness, that in the resonance case the theory will actually give the result it ought to give, in order to agree with Wood’s resonance phenomenon: an increase of the proper vibration un, which causes the crisis, to a finite magnitude comparable with that of the originally existing uk , from which, of course, “spontaneous emission” of the spectral line (uk un) results. I do not wish, however, to attempt to work out the calculation of the resonance case fully here, because the result would be of little value, so long as the reaction of the emitted radiation on the emitting system is not taken into account. Such a reaction must exist, not only because there is no ground at all for differentiating on principle between the light wave which is incident from outside, and that which is emitted by the system itself, but also because otherwise, if several proper vibrations were simultaneously excited in a system left to itself, the spontaneous emission would continue indefinitely. This required back-coupling must act so that in this case, along with the light emission, the higher proper vibrations gradually die down, and, finally, the fundamental vibration, corresponding to the normal state of the system, alone remains. The back-coupling is evidently exactly analogous to the reaction of radiation in the classical electron theory. This analogy also allays the increasing apprehension caused by the previous neglect of this back-coupling. The influence of the relevant term (probably no longer linear) in the wave equation will generally be small, just as in the electron the back pressure of radiation is generally very small compared with the force of inertia and the external field strength. In the resonance case, however—just as in the electron theory—the coupling with the proper light wave will be of the same order as that with the incident wave, and must be taken into account, if the “equilibrium” between the different proper vibrations, which sets in for the given irradiation, is to be correctly computed.
The Dreams That Stuff is Made of Page 35