Hr. W. Pauli, jun., informs me that he has arrived at the following closed formulae for the total intensity of the lines in the Lyman and Balmer series, through a modification of the method given in section 2 of the Appendix. For the Lyman series these are
and for the Balmer series
The total emission intensities (square of amplitudes into fourth power of the frequency) are proportional to these expressions, within the series in question. The numbers obtained from the formula for the Balmer series are in complete agreement with those given on pp. 404, 405.
Zürich, Physical Institute of the University.
(Received May 10, 1926.)
QUANTISATION AS A PROBLEM OF PROPER VALUES (PART IVfo)
(Annalen der Physik (4), vol. 81, 1926)
ABSTRACT: § 1. Elimination of the energy-parameter from the vibration equation. The real wave equation. Non-conservative systems. §2. Extension of the perturbation theory to perturbations which explicitly contain the time. Theory of dispersion. § 3. Supplementing § 2. Excited atoms, degenerate systems, continuous spectrum. § 4. Discussion of the resonance case. § 5. Generalisation for an arbitrary perturbation. § 6. Relativistic-magnetic generalisation of the fundamental equations. § 7. On the physical significance of the field scalar.
§ 1. ELIMINATION OF THE ENERGY-PARAMETER FROM THE VIBRATION EQUATION. THE REAL WAVE EQUATION. NON-CONSERVATIVE SYSTEMS
The wave equation (18) or (18”) of Part II., viz.
(1)
or
(1ʹ)
which forms the basis for the re-establishment of mechanics attempted in this series of papers, suffers from the disadvantage that it expresses the law of variation of the “mechanical field scalar” ψ , neither uniformly nor generally. Equation (1) contains the energy- or frequency-parameter E, and is valid, as is expressly emphasized in Part II., with a definite E-value inserted, for processes which depend on the time exclusively through a definite periodic factor:
(2)
Equation (1) is thus not really any more general than equation (1ʹ), which takes account of the circumstance just mentioned and does not contain the time at all.
Thus, when we designated equation (1) or (1ʹ), on various occasions, as “the wave equation”, we were really wrong and would have been more correct if we had called it a “vibration-” or an “amplitude-”equation. However, we found it sufficient, because to it is linked the Sturm-Liouville proper value problem—just as in the mathematically strictly analogous problem of the free vibrations of strings and membranes—and not to the real wave equation.
As to this, we have always postulated up till now that the potential energy V is a pure function of the co-ordinates and does not depend explicitly on the time. There arises, however, an urgent need for the extension of the theory to non-conservative systems, because it is only in that way that we can study the behaviour of a system under the influence of prescribed external forces, e.g. a light wave, or a strange atom flying past. Whenever V contains the time explicitly, it is manifestly impossible that equation (1) or (1ʹ) should be satisfied by a function ψ , the method of dependence of which on the time is as given by (2). We then find that the amplitude equation is no longer sufficient and that we must search for the real wave equation.
For conservative systems, the latter is easily obtained. (2) is equivalent to
(3)
We can eliminate E from (1ʹ) and (3) by differentiation, and obtain the following equation, which is written in a symbolic manner, easy to understand:
(4)
This equation must be satisfied by every ψ which depends on the time as in (2), though with E arbitrary, and consequently also by every ψ which can be expanded in a Fourier series with respect to the time (naturally with functions of the co-ordinates as coefficients). Equation (4) is thus evidently the uniform and general wave equation for the field scalar ψ .
It is evidently no longer of the simple type arising for vibrating membranes, but is of the fourth order, and of a type similar to that occurring in many problems in the theory of elasticity.fp However, we need not fear any excessive complication of the theory, or any necessity to revise the previous methods, associated with equation (1ʹ). If V does not contain the time, we can, proceeding from (4), apply (2), and then split up the operator as follows:
(4’)
By way of trial, we can resolve this equation into two “alternative” equations, namely, into equation (1ʹ) and into another, which only differs from (1′) in that its proper value parameter will be called minus E, instead of plus E. According to (2) this does not lead to new solutions. The decomposition of (4’) is not absolutely cogent, for the theorem that “a product can only vanish when at least one factor vanishes” is not valid for operators. This lack of cogency, however, is a feature common to all the methods of solution of partial differential equations. The procedure finds its subsequent justification in the fact that we can prove the completeness of the discovered proper functions, as functions of the co-ordinates. This completeness, coupled with the fact that the imaginary part as well as the real part of (2) satisfies equation (4), allows arbitrary initial conditions to be fulfilled by ψ and ∂ ψ /∂ t.
Thus we see that the wave equation (4), which contains in itself the law of dispersion, can really stand as the basis of the theory previously developed for conservative systems. The generalisation for the case of a time-varying potential function nevertheless demands caution, because terms with time derivatives of V may then appear, about which no information can be given to us by equation (4), owing to the way we obtained it. In actual fact, if we attempt to apply equation (4) as it stands to non-conservative systems, we meet with complications, which seem to arise from the term in ∂ V/∂ t. Therefore, in the following discussions, I have taken a somewhat different route, which is much easier for calculations, and which I consider is justified in principle.
We need not raise the order of the wave equation to four, in order to get rid of the energy-parameter. The dependence of ψ on the time, which must exist if (1′) is to hold, can be expressed by
(3′)
as well as by (3). We thus arrive at one of the two equations
(4”)
We will require the complex wave function ψ to satisfy one of these two equations. Since the conjugate complex function ψ will then satisfy the other equation, we may take the real part of ψ as the real wave function (if we require it). In the case of a conservative system (4”) is essentially equivalent to (4), as the real operator may be split up into the product of the two conjugate complex operators if V does not contain the time.
§ 2. EXTENSION OF THE PERTURBATION THEORY TO PERTURBATIONS CONTAINING THE TIME EXPLICITLY. THEORY OF DISPERSION
Our main interest is not in systems for which the time and spatial variations of the potential energy V are of the same order of magnitude, but in systems, conservative in themselves, which are perturbed by the addition of small given functions of the time (and of the co-ordinates) to the potential energy. Let us, therefore, write
(5)
where, as often before, x represents the whole of the configuration coordinates. We regard the unperturbed proper value problem (r = 0) as solved. Then the perturbation problem can be solved by quadratures.
However, we will not treat the general problem immediately, but will select the problem of the dispersion theory out of the vast number of weighty applications which fall under this heading, on account of its striking importance, which really justifies a separate treatment in any case. Here the perturbing forces originate in an alternating electric field, homogeneous and vibrating synchronously in the domain of the atom; and thus, if we have to do with a linearly polarised monochromatic light of frequency v, we write
(6)
and hence
(5′)
Here A(x ) is the negative product of the light-amplitude and the coordinate function which, according to ordinary mechanics, signifies the component of the electric moment of the atom in the di
rection of the electric light-vector (say − FΣe i zi , if F is the light-amplitude, ei , zi the charges and z-co-ordinates of the particles, and the light is polarised in the z-direction). We borrow the time-variable part of the potential function from ordinary mechanics with just as much or as little right as previously, e.g. in the Kepler problem, we borrowed the constant part.
Using (5′), equation (4”) becomes
(7)
For A = 0, these equations are changed by the substitution
(8)
(which is now to be taken in the literal sense, and does not imply pars realis) into the amplitude equation (1ʹ) of the unperturbed problem, and we know (cf. § 1) that the totality of the solutions of the unperturbed problem is found in this way. LetEk and uk (x); k = 1, 2, 3, . . .
be the proper values and normalised proper functions of the unperturbed problem, which we regard as known, and which we will assume to be discrete and different from one another (non-degenerate system with no continuous spectrum), so that we may not become involved in secondary questions, requiring special consideration.
Just as in the case of a perturbing potential independent of the time, we will have to seek solutions of the perturbed problem in the neighbourhood of each possible solution of the unperturbed problem, and thus in the neighbourhood of an arbitrary linear combination of the uk ’s, which has constant co-efficients [from (8), the uk ’s to be combined with the appropriate time factors . The solution of the perturbed problem, lying in the neighbourhood of a definite linear combination, will have the following physical meaning. It will be this solution which first appears, if, when the light wave arrived, precisely that definite linear combination of free proper vibrations was present (perhaps with trifling changes during the “excitation”).
Since, however, the equation of the perturbed problem is also homogeneous—let this want of analogy with the “forced vibrations” of acoustics be expressly emphasized—it is evidently sufficient to seek the perturbed solution in the neighbourhood of each separate
(9)
as we may then linearly combine these ad lib., just as for unperturbed solutions.
To solve the first of equations (7) we therefore now put
(10)
[The lower symbol, i.e. the second of equations (7), is henceforth left on one side, as it would not yield anything new.] The additional term w(x , t) can be regarded as small, and its product with the perturbing potential neglected. Bearing this in mind while substituting from (10) in (7), and remembering that uk (x) and Ek are proper functions and values of the unperturbed problem, we get
(11)
This equation is readily, and really only, satisfied by the substitution
(12)
where the two functions w ± respectively obey the two equations
(13)
This step is essentially unique. At first sight, we apparently can add to (12) an arbitrary aggregate of unperturbed proper vibrations. But this aggregate would necessarily be assumed small, of the first order (since this has been assumed for w), and thus does not interest us at present, as it could only produce perturbations of the second order at most.
In equations (13) we have at last those non-homogeneous equations we might have expected to encounter—in spite of the lack of analogy with real forced vibrations, as emphasized above. This lack of analogy is extraordinarily important and manifests itself in equations (13) in the two following particulars. Firstly, as the “second member” (“exciting force”), the perturbation function A(x ) does not appear alone, but multiplied by the amplitude of the free vibration already present. This is indispensable if the physical facts are to be properly taken into account, for the reaction of an atom to an incident light wave depends almost entirely on the state of the atom at that time, whereas the forced vibrations of a membrane, plate, etc., are known to be quite independent of the proper vibrations which may be superimposed on them, and thus would yield an obviously wrong representation of our case. Secondly, in place of the proper value on the left-hand side of (13), i.e. as “exciting frequency”, we do not find the frequency ν of the perturbing force alone, but rather in one case added to, and in the other subtracted from, that of the free vibration already present. This is equally indispensable. Otherwise the proper frequencies themselves , which correspond to the term-frequencies, would function as resonance-points, and not the differences of the proper frequencies, as is demanded, and is really given by equation (13). Moreover, we see with satisfaction that the latter gives only the differences between a proper frequency which is actually excited and all the others, and not the differences between pairs of proper frequencies, of which no member is excited.
In order to investigate this more closely, let us complete the solution. By well-known methodsfqwe find, as simple solutions of (13),
(14)
where
(15)
ρ(x) is the “density function”, i.e. that function of the position-coordinates with which equation (1ʹ) must be multiplied to make it self-adjoint. The un(x )’s are assumed to be normalised. It is further postulated that hν does not agree exactly with any of the differences Ek−En of the proper values. This “resonance case” will be dealt with later (cf. § 4).
If we now form from (14), using (12) and (10), the entire perturbed vibration, we get
(16)
Thus in the perturbed case, along with each free vibration uk (x ) occur in small amplitude all those vibrations un(x), for which a′kn ≠ 0. The latter are exactly those, which, if they exist as free vibrations along with uk , give rise to a radiation, which is (wholly or partially) polarised in the direction of polarisation of the incident wave. For apart from a factor, a′kn is just the component amplitude, in this direction of polarisation, of the atom’s electric moment, which is oscillating with frequency (Ek − En )/ h, according to wave mechanics, and which appears when u k and u n exist together.fr The simultaneous oscillation, however, takes place with neither the proper frequency En/h, peculiar to these vibrations, nor the frequency v of the light wave, but rather with the sum and difference of ν and Ek/h (i. e. the frequency of the one existing free vibration).
The real or the imaginary part of (16) can be considered as the real solution. In the following, however, we will operate with the complex solution itself.
To see the significance that our result has in the theory of dispersion, we must examine the radiation arising from the simultaneous existence of the excited forced vibrations and the free vibration, already present. For this purpose, we form, following the method wefs have always adopted above—a criticism follows in § 7—the product of the complex wave function (16) and its conjugate, i.e. the norm of the complex wave function ψ . We notice that the perturbing terms are small, so that squares and products may be neglected. After a simple reductionft we obtain
(17)
According to the heuristic hypothesis on the electrodynamical significance of the field scalar ψ, the present quantity—apart from a multiplicative constant—represents the electrical density as a function of the space co-ordinates and the time, if x stands for only three space co-ordinates, i.e. if we are dealing with the problem of one electron. We remember that the same hypothesis led us to correct selection and polarisation rules and to a very satisfactory representation of intensity relationships in our discussion of the hydrogen Stark effect. By a natural generalisation of this hypothesis—of which more in § 7—we regard the following as representing in the general case the density of the electricity, which is “associated” with one of the particles of classical mechanics, or which “originates in it”, or which “corresponds to it in wave mechanics”: the integral of ψψ taken over all those co-ordinates of the system, which in classical mechanics fix the position of the rest of the particles, multiplied by a certain constant, the classical “charge” of the first particle. The resultant density of charge at any point of space is then represented by the sum of such integrals taken over all the particles.
Thus in order
to find any space component whatever of the total wave-mechanical dipole moment as a function of the time, we must, on this hypothesis, multiply expression (17) by that function of the co-ordinates which gives that particular dipole - component in classical mechanics as a function of the configuration of the point system, e.g. by
(18)
My = Σeiyi if we are dealing with the dipole moment in the y-direction. Then we have to integrate over all the configuration co-ordinates.
Let us work this out, using the abbreviation
(19)
Let us elucidate further the definition (15) of the ’s by recalling that if the incident electric light-vector is given by
The Dreams That Stuff is Made of Page 34