(34ʹ)
This definition is different from all previous statements (except perhaps that of Heisenberg?). Yet, from various arguments from experiment we were led to put “half-integral” values for n in formula (31). It is easily seen that (34ʹ) gives practically the same as (31) with half-integral values of n. For
The discrepancy consists only of a small additive constant; the level differences in (34ʹ) are the same as are got from “half-integral quantisation”. This is true also for the application to short-wave bands, where the moment of inertia is not the same in the initial and final states, on account of the “electronic jump”. For at most a small constant additional part comes in for all lines of a band, which is swamped in the large “electronic term” or in the “nuclear vibration term”. Moreover, our previous analysis does not permit us to speak of this small part in any more definite way than as, say,
The notion of the moment of inertia being fixed by “quantun conditions” for electronic motions and nuclear vibrations follows naturally from the whole line of thought developed here. We will show in the next section how we can treat, approximately at least, the nuclear vibrations and the rotations of the diatomic molecule simultaneously by a synthesis of the casesea considered in 1 and 3.
I should like to mention also that the value n = 0 corresponds not to the vanishing of the wave function ψ but to a constant value for it, and accordingly to a vibration with amplitude constant over the whole sphere.
4. Non-rigid Rotator (Diatomic Molecule)
According to the observation at the end of section 2, we must state the problem initially with all the six degrees of freedom that the rotator really possesses. Choose Cartesian co-ordinates for the two molecules, viz. x1, y1, z1; x2, y2, z2, and let the masses be m1 and m2, and r be their distance apart. The potential energy is
(35)
where
Here
(36)
may be called the “resultant mass”. Then ν0 is the mechanical proper frequency of the nuclear vibration, regarding the line joining the nuclei as fixed, and r0 is the distance apart for which the potential energy is a minimum. These definitions are all in the sense of the usual mechanics.
For the vibration equation (18”) we get the following:
(37)
Introduce new independent variables x, y, z, ξ, η, ζ, where
(38)
The substitution gives
(37ʹ)
where for brevity
(39)
Now we can put for ψ the product of a function of the relative coordinates x, y, z, and a function of the co-ordinates of the centre of mass ξ, η, ζ:
(40)
For g we get the defining equation
(41)
This is of the same form as the equation for the motion, under no forces, of a particle of mass m1 + m2. The constant would in this case have the meaning
(42)
where Et is the energy of translation of the said particle. Imagine this value inserted in (41). The question as to the values of Et admissible as proper values depends now on this, whether the whole infinite space is available for the original co-ordinates and hence for those of the centre of gravity without new potential energies coming in, or not. In the first case every non-negative value is permissible and every negative value not permissible. For when Et is not negative and only then, (41) possesses solutions which do not vanish identically and yet remain finite in all space. If, however, the molecule is situated in a “vessel”, then the latter must supply boundary conditions for the function g, or in other words, equation (41), on account of the introduction of further potential energies, will alter its form very abruptly at the walls of the vessel, and thus a discrete set of Et -values will be selected as proper values. It is a question of the “Quantisation of the motion of translation”, the main points of which I have lately discussed, showing that it leads to Einstein’s Gas Theory.eb
For the factor f of the vibration function ψ, depending on the relative co-ordinates x, y, z, we get the denning equation
(43)
where for brevity we put
(39ʹ)
We now introduce instead of x, y, z, the spherical polars r, θ, φ (which is in agreement with the previous use of r). After multiplying by µ we get
(43ʹ)
Now break up f. The factor depending on the angles is a surface harmonic. Let the order be n. The curled bracket is −n(n + 1)f. Imagine this inserted and for simplicity let f now stand for the factor depending on r. Then introduce as new dependent variable
(44)
and as new independent variable
(45)
The substitution gives
(46)
To this point the analysis has been exact. Now we will make an approximation, which I well know requires a stricter justification than I will give here. Compare (46) with equation (22ʹ) treated earlier. They agree in form and only differ in the coefficient of the unknown function by terms of the relative order of magnitude of . This is seen, if we develop thus:
(47)
substitute in (46), and arrange in powers of ρ/r0. If we introduce for ρ a new variable differing only by a small constant, viz.
(48)
then equation (46) takes the form
(46ʹ)
where we have put
(49)
The symbol in (46ʹ) represents terms which are small compared with the retained term of the order of .
Now we know that the first proper functions of equation (22ʹ), to which we now compare (46’), only differ markedly from zero in a small range on both sides of the origin. Only those of higher order stretch gradually further out. For moderate orders, the domain for equation (46ʹ), if we neglect the term and bear in mind the order of magnitude of molecular constants, is indeed small compared with r0. We thus conclude (without rigorous proof, I repeat), that we can in this way obtain a useful approximation for the first proper functions, within the region where they differ at all markedly from zero, and also for the first proper values. From the proper value condition (25) and omitting the abbreviations (49), (39’), and (39), though introducing the small quantity
(50)
instead, we can easily derive the following energy steps,
(51)
where
(52)
is still written for the moment of inertia.
In the language of classical mechanics, ε is the square of the ratio of the frequency of rotation to the vibration frequency ν0; it is therefore rlally a small quantity in the application to the molecule, and formula (51) has the usual structure, apart from this small correction and the other differences already mentioned. It is the synthesis of (25′) and (34ʹ) to which Et is added as representing the energy of translation. It must be emphasized that the value of the approximation is to be judged not only by the smallness of ε but also by l not being too large. Practically, however, only small numbers have to be considered for l.
The ε-corrections in (51) do not yet take account of deviations of the nuclear vibrations from the pure harmonic type. Thus a comparison with Kratzer’s formula (vide Sommerfeld, loc. cit.) and with experience is impossible. I only desired to mention the case provisionally, as an example showing that the intuitive idea of the equilibrium configuration of the nuclear system retains its meaning in undulatory mechanics also, and showing the manner in which it does so, provided that the wave amplitude ψ is different from zero practically only in a small neighbourhood of the equilibrium configuration. The direct interpretation of this wave function of six variables in three-dimensional space meets, at any rate initially, with difficulties of an abstract nature.
The rotation-vibration-problem of the diatomic molecule will have to be re-attacked presently, the non-harmonic terms in the energy of binding being taken into account. The method, selected skilfully by Kratzer for the classical mechanical treatment, is also suitable for undulatory mechanics. If, however, we are going to push the calculation as far as is nece
ssary for the fineness of band structure, then we must make use of the theory of the perturbation of proper values and functions, that is, of the alteration experienced by a definit proper value and the appertaining proper functions of a differential equation, when there is added to the coefficient of the unknown function in the equation a small “disturbing term”. This “perturbation theory” is the complete counterpart of that of classical mechanics, except that it is simpler because in undulatory mechanics we are always in the domain of linear relations. As a first approximation we have the statement that the perturbation of the proper value is equal to the perturbing term averaged “over the undistrubed motion”.
The perturbation theory broadens the analytical range of the new theory extraordinarily. As an important practical success, let me say here that the Stark effect of the first order will be found to be really completely in accord with Epstein’s formula, which has become unimpeachable through the confirmation of experience.
Zürich, Physical Institute of the University.
(Received February 23, 1926.)
QUANTISATION AS A PROBLEM OF PROPER VALUES (PART III)
PERTURBATION THEORY, WITH APPLICATION TO THE STARK EFFECT OF THE BALMER LINES
(Annalen der Physik (4), vol. 80, 1926)
INTRODUCTION. ABSTRACT
As has already been mentioned at the end of the preceding paper,ec the available range of application of the proper value theory can by comparatively elementary methods be considerably increased beyond the “directly soluble problems”; for proper values and functions can readily be approximately determined for such boundary value problems as are sufficiently closely related to a directly soluble problem. In analogy with ordinary mechanics, let us call the method in question the perturbation method. It is based upon the important property of continuity possessed by proper values and functions,ed principally, for our purpose, upon their continuous dependence on the coefficients of the differential equation, and less upon the extent of the domain and on the boundary conditions, since in our case the domain (“entire q-space”) and the boundary conditions (“remaining finite”) are generally the same for the unperturbed and perturbed problems.
The method is essentially the same as that used by Lord Rayleigh in investigatingee the vibrations of a string with small inhomogeneities in his Theory of Sound (2nd edit., vol. i., pp. 115–118, London, 1894). This was a particularly simple case, as the differential equation of the unperturbed problem had constant coefficients, and only the perturbing terms were arbitrary functions along the string. A complete generalisation is possible not merely with regard to these points, but also for the specially important case of several independent variables, i.e. for partial differential equations, in which multiple proper values appear in the unperturbed problem, and where the addition of a perturbing term causes the splitting up of such values and is of the greatest interest in well-known spectroscopic questions (Zeeman effect, Stark effect, Multiplicities). In the development of the perturbation theory in the following Section I., which really yields nothing new to the mathematician, I put less value on generalising to the widest possible extent than on bringing forward the very simple rudiments in the clearest possible manner. From the latter, any desired generalisation arises almost automatically when needed. In Section II., as an example, the Stark effect is discussed and, indeed, by two methods, of which the first is analogous to Epstein’s method, by which he first solvedef the problem on the basis of classical mechanics, supplemented by quantum conditions, while the second, which is much more general, is analogous to the method of secular perturbations.eg The first method will be utilised to show that in wave mechanics also the perturbed problem can be “separated” in parabolic co-ordinates, and the perturbation theory will first be applied to the ordinary differential equations into which the original vibration equation is split up. The theory thus merely takes over the task which on the old theory devolved on Sommerfeld’s elegant complex integration for the calculation of the quantum integrals.eh In the second method, it is found that in the case of the Stark effect an exact separation co-ordinate system exists, quite by accident, for the perturbed problem also, and the perturbation theory is applied directly to the partial differential equation. This latter proceeding proves to be more troublesome in wave mechanics, although it is theoretically superior, being more capable of generalisation.
Also the problem of the intensity of the components in the Stark effect will be shortly discussed in Section II. Tables will be calculated, which, as a whole, agree even better with experiment than the well-known ones calculated by Kramers with the help of the correspondence principle.ei
The application (not yet completed) to the Zeeman effect will naturally be of much greater interest. It seems to be indissolubly linked with a correct formulation in the language of wave mechanics of the relativistic problem, because in the four-dimensional formulation the vector-potential automatically ranks equally with the scalar. It was already mentioned in Part I. that the relativistic hydrogen atom may indeed be treated without further discussion, but that it leads to “half-integral” azimuthal quanta, and thus contradicts experience. Therefore “something must still be missing”. Since then I have learnt what is lacking from the most important publications of G. E. Uhlenbeck and S. Goudsmit,ej and then from oral and written communications from Paris (P. Langevin) and Copenhagen (W. Pauli), viz., in the language of the theory of electronic orbits, the angular momentum of the electron round its axis, which gives it a magnetic moment. The utterances of these investigators, together with two highly significant papers by Slaterek and by Sommerfeld and Unsöldel dealing with the Balmer spectrum, leave no doubt that, by the introduction of the paradoxical yet happy conception of the spinning electron, the orbital theory will be able to master the disquieting difficulties which have latterly begun to accumulate (anomalous Zeeman effect; Paschen-Back effect of the Balmer lines; irregular and regular Röntgen doublets; analogy of the latter with the alkali doublets, etc.). We shall be obliged to attempt to take over the idea of Uhlenbeck and Goudsmit into wave mechanics. I believe that the latter is a very fertile soil for this idea, since in it the electron is not considered as a point charge, but as continuously flowing through space,em and so the unpleasing conception of a “rotating point-charge” is avoided. In the present paper, however, the taking over of the idea is not yet attempted.
To the third section, as “mathematical appendix”, have been relegated numerous uninteresting calculations—mainly quadratures of products of proper functions, required in the second section. The formulae of the appendix are numbered (101), (102), etc.
I. PERTURBATION THEORY
§ 1. A SINGLE INDEPENDENT VARIABLE
Let us consider a linear, homogeneous, differential expression of the second order, which we may assume to be in self-adjoint form without loss of generality, viz.
(1)
y is the dependent function; p , p′ and q are continuous functions of the independent variable x and p ≥ 0. A dash denotes differentiation with respect to x (p′ is therefore the derivative of p, which is the condition for self-adjointness).
Now let p(x) be another continuous function of x, which never becomes negative, and also in general does not vanish. We consider the proper value problem of Sturm and Liouville,en
(2)
It is a question, first, of finding all those values of the constant E (“proper values”) for which the equation (2) possesses solutions y(x), which are continuous and not identically vanishing within a certain domain, and which satisfy certain “boundary conditions” at the bounding points; and secondly of finding these solutions (“proper functions”) themselves. In the cases treated in atomic mechanics, domain and boundary conditions are always “natural”. The domain, for example, reaches from 0 to ∞, when x signifies the value of the radius vector or of an intrinsically positive parabolic co-ordinate, and the boundary conditions are in these cases: remaining finite. Or, when x signifies an azimuth, then the domain is the
interval from 0 to 2π and the condition is: Repetition of the initial values of y and y′ at the end of the interval (“periodicity”).
It is only in the case of the periodic condition that multiple, viz. double-valued, proper values appear for one independent variable. By this we understand that to the same proper value belong several (in the particular case, two) linearly independent proper functions. We will now exclude this case for the sake of simplicity, as it attaches itself easily to the developments of the following paragraph. Moreover, to lighten the formulae, we will not expressly take into account in the notation the possibility that a “band spectrum” (i.e. a continuum of proper values) may be present when the domain extends to infinity.
The Dreams That Stuff is Made of Page 29