We again fix our attention on a definite proper value Ek. Let (16) be a system of proper functions belonging to it, which we assume to be normalised and orthogonal to one another in the sense described above, but not yet fitted to the particular perturbation in the sense explained, because to find the substitution that leads to this fitting is precisely our chief task! In place of (5), § 1, we must now put for the perturbed quantities the following,
(18)
wherein the vl(x)’s are functions, and the εl’s and the kli’s are systems of constants, which are still to be defined, but which we initially do not limit in any way, although we know that the system of coefficients kli muster form an orthogonal substitution. The index k should still be attached to the three types of quantity named, in order to indicate that the whole discussion refers to the kth proper value of the unperturbed problem. We refrain from carrying this out, in order to avoid the confusing accumulation of indices. The index k is to be assumed fixed in the whole of the following discussion, until the contrary is stated.
Let us select one of the perturbed proper functions and values by giving a definite value to the index l in (18), and let us substitute from (18) in the differential equation (2ʹ) and arrange in powers of λ. Then the terms independent of λ disappear exactly as in § 1, because the unperturbed proper quantities satisfy equation (2), by hypothesis. Only terms containing the first power of λ remain, as we can strike out the others. Omitting a factor λ, we get
(19)
and thus obtain again for the definition of the perturbation vl of the functions a non-homogeneous equation, to which corresponds as homogeneous equation the equation (2), with the particular value E = Ek, i.e. the equation satisfied by the set of functions uki;i = 1, 2, ... α. The form of the left side of equation (19) is independent of the index l.
On the right side occur εl and kli, the constants to be defined, and we are thus enabled to evaluate them, even before calculating vl. For, in order that (19) should have a solution at all, it is necessary and sufficient that its right-hand side should be orthogonal to all the proper functions of the homogeneous equation (2) belonging to Ek. Therefore, we must have
(20)
i.e. on account of the normalisation (17),
(21)
If we write, briefly, for the symmetrical matrix of constants, which can be evaluated by quadrature,
(22)
then we recognise in
(21′)
a system of α linear homogeneous equations for the calculation of the α constants klm; m = 1, 2 . . . α, where the perturbation εl of the proper value still occurs in the coefficients, and is itself unknown. However, this serves for the calculation of εl before that of the klm ’s. For it is known that the linear homogeneous system (21ʹ) of equations has solutions if, and only if, its determinant vanishes. This yields the following algebraic equation of degree α for εl:
(23)
We see that the problem is completely identical with the transformation of the quadratic form in α variables, with coefficients εmi, to its principal axes. The “secular equation” (23) yields α roots for εl, the “reciprocal of the squares of the principal axes”, which in general are different, and on account of the symmetry of the εmi’s always real. We thus get all the α perturbations of the proper values (l = 1, 2 . . . α) at the same time, and would have inferred the splitting up of an α-fold proper value into exactly α simple values, generally different, even had we not assumed it already, as fairly obvious. For each of these εl -values, equations (21ʹ) give a system of quantities kli i = 1, 2, . . . α, and, as is known, only one (apart from a general constant factor), provided all the εl’s are really different. Further, it is known that the whole system of α2 quantities kli forms an orthogonal system of coefficients, defining as usual, in the principal axes problem, the directions of the new coordinate axes with reference to the old ones. We may, and will, employ the undefined factors just mentioned to normalise the kli’s completely as “direction cosines”, and this, as is easily seen, makes the perturbed proper functions turn out normalised again, according to (18), at least in the “zero approximation” (i.e. apart from the λ-terms).
If the equation (23) has multiple roots, then we have the case previously mentioned, when the perturbation does not completely remove the degeneration. The perturbed equation has then multiple proper values also and the definition of the constants kli becomes partially arbitrary. This has no consequence other than that (as is always the case with multiple proper values) we must and may acquiesce, even after the perturbation is applied, in a system of proper functions which in many respects is still arbitrary.
The main task is accomplished with this transformation to principal axes, and we will often find it sufficient in the applications in quantum theory to define the proper values to a first and the functions to zero approximation. The evaluation of the constants kli and εli cannot be carried out always, since it depends on the solution of an algebraic equation of degree α. At the worst there are methodses which give the evaluation to any desired approximation by a rational process. We may thus regard these constants as known, and will now give the calculation of the functions to the first approximation, for the sake of completeness. The procedure is exactly as in §1.
We have to solve equation (19) and to that end we write vl as a series of the whole set of proper functions of (2),
(24)
The summation is to extend with respect to k′ from 0 to ∞, and, for each fixed value of kʹ, for i′ varying over the finite number of proper functions which belong to Ek′. (Now, for the first time, we take account of proper functions which do not belong to the α-fold value Ek we are fixing our attention on.) Secondly, we develop the right-hand side of (19), divided by ρ(x), in a series of the entire set of proper functions,
(25)
wherein
(26)
(the last two equalities follow from (17) and (20) respectively). On substituting from (24) and (25) in (19), we get
(27)
Since uk′i′ satisfies equation (2) with E = Ek, this gives
(28)
By equating coefficients on right and left, all the γl,k′i′’s are defined, with the exception of those in which k′ = k. Thus
(29)
while those γ’s for which k′ = k are of course not fixed by equation (19). This again corresponds to the fact that we have provisionally normalised the perturbed functions , of (18), only in the zero approximation (through the normalisation of the κli’s), and it is easily recognised again that we have to put the whole of the γ -quantities in question equal to zero, in order to bring about the normalisation of the even in the first approximation. By substituting from (29) in (24), and then from (24) in (18), we finally obtain for the perturbed proper functions to a first approximation
(30)
The dash on the second sigma indicates that all the terms with k′ = k are to be omitted. In the application of the formula for an arbitrary k, it is to be observed that the κli’s, as obviously also the multiplicity α of the proper value Ek, to which we have specially directed our attention, still depend on the index k, though this is not expressed in the symbols. Let us repeat here that the κli’s are to be calculated as a system of solutions of equations (21′), normalised so that the sum of the squares is unity, where the coefficients of the equations are given by (22), while for the quantity εl in (21′), one of the roots of (23) is to be taken. This root then gives the allied perturbed proper value, from
(31)
Formulae (30) and (31) are the generalisations of (14) and (15) of § 1.
It need scarcely be said that the extensions and generalisations mentioned at the end of § 1 can of course take effect here also. It is hardly worth the trouble to carry out these developments generally. We succeed best in any special case if we do not use ready-made formulae, but go directly by the simple fundamental principles, which have been explained, perhaps too minutely, in the present paper. I would only like to consider
briefly the possibility, already mentioned at the end of § 1, that the equation (2) perhaps may lose (and indeed in the case of several variables irreparably lose), its self-adjoint character if the perturbing terms also contain derivatives of the unknown function. From general theorems we know that then the proper values of the perturbed equation no longer need to be real. We can illustrate this further. We can easily see, by carrying out the developments of this paragraph, that the elements of determinant (23) are no longer symmetrical, when the perturbing term contains derivatives. It is known that in this case the roots of equation (23) no longer require to be real.
The necessity for the expansion of certain functions in a series of proper functions, in order to arrive at the first or zero approximation of the proper values or functions, can become very inconvenient, and can at least complicate the calculation considerably in cases where an extended spectrum co-exists with the point spectrum and where the point spectrum has a limiting point (point of accumulation) at a finite distance. This is just the case in the problems appearing in the quantum theory. Fortunately it is often—perhaps always—possible, for the purpose of the perturbation theory, to free oneself from the generally very troublesome extended spectrum, and to develop the perturbation theory from an equation which does not possess such a spectrum, and whose proper values do not accumulate near a finite value, but grow beyond all limits with increasing index. We will become acquainted with an example in the next paragraph. Of course, this simplification is only possible when we are not interested in a proper value of the extended spectrum.
II. APPLICATION TO THE STARK EFFECT
§ 3. CALCULATION OF FREQUENCIES BY THE METHOD WHICH CORRESPONDS TO THAT OF EPSTEIN
If we add a potential energy +e F z to the wave equation (5), Part I. (p. 2), of the Kepler problem, corresponding to the influence of an electric field of strength F in the positive z-direction, on a negative electron of charge e, then we obtain the following wave equation for the Stark effect of the hydrogen atom,
(32)
which forms the basis of the remainder of this paper. In § 5 we will apply the general perturbation theory of §2 directly to this partial differential equation. Now, however, we will lighten our task by introducing space parabolic co-ordinates λ1 λ2, φ, by the following equations,
(33)
λ1 and λ2 run from 0 to infinity; the corresponding co-ordinate surfaces are the two sets of confocal paraboloids of revolution, which have the origin as focus and the positive (λ2) or negative (λ1) z-axis respectively as axes. φ runs from 0 to 2π , and the co-ordinate surfaces belonging to it are the set of half planes limited by the z-axis. The relation of the co-ordinates is unique. For the functional determinant we get
(34)
The space element is thus
(35)
We notice, as consequences of (33),
(36)
The expression of (32) in the chosen co-ordinates gives, if we multiply by (34)et (to restore the self-adjoint form),
(32ʹ)
Here we can again take—and this is the why and wherefore of all “methods” of solving linear partial differential equations—the function ψ as the product of three functions, thus,
(37)
each of which depends on only one co-ordinate. For these functions we get the ordinary differential equations
(38)
wherein n and β are two further “proper value-like” constants of integration (in addition to E), still to be defined. By the choice of symbol for the first of these, we have taken into account the fact that the first of equations (38) makes it take integral values, if Φ and are to be continuous and single-valued functions of the azimuth φ. We then have
(39)
and it is evidently sufficient if we do not consider negative values of n. Thus
(40)
In the symbol used for the second constant β , we follow Sommerfeld (Atombau, 4th edit., p. 821) in order to make comparison easier. (Similarly, below, with A, B, C, D.) We treat the last two equations of (38) together, in the form
(41)
where
(42)
and the upper sign is valid for Λ = Λ1, ξ = λ1, and the lower one for Λ = Λ2, ξ = λ2. (Unfortunately, we have to write ξ instead of the more appropriate λ, to avoid confusion with the perturbation parameter λ of the general theory, §§ and 2.)
If we omit initially in (41) the Stark effect term Dξ 2, which we conceive as a perturbing term (limiting case for vanishing field), then this equation has the same general structure as equation (7) of Part I., and the domain is also the same, from 0 to ∞. The discussion is almost the same, word for word, and shows that non-vanishing solutions, which, with their derivatives, are continuous and remain finite within the domain, only exist if either A > 0 (extended spectrum, corresponding to hyperbolic orbits) or
(43)
If we apply this to the last two equations of (38) and distinguish the two k-values by suffixes 1 and 2, we obtain
(44)
By addition, squaring and use of (42) we find
(45)
These are the well-known Balmer-Bohr elliptic levels, where as principal quantum number enters
(46)
We get the discrete term spectrum and the allied proper functions in a way simpler than that indicated, if we apply results already known in mathematical literature as follows. We transform first the dependent variable λ in (41) by putting
(47)
and then the independent ξ by putting
(48)
We find for u as a function of η the equation
(41ʹ)
This equation is very intimately connected with the polynomials named after Laguerre. In the mathematical appendix, it will be shown that the product of and the nth derivative of the (n + k )th Laguerre polynomial satisfies the differential equation
(103)
and that, for a fixed n, the functions named form the complete system of proper functions of the equation just written, when k runs through all non-negative integral values. Thus it follows that, for vanishing D, equation (41ʹ) possesses the proper functions
(49)
and the proper values
(50)
—and no others! (See the mathematical appendix concerning the remarkable loss of the extended spectrum caused by the apparently inoffensive transformation (48); by this loss the development of the perturbation theory is made much easier.)
We have now to calculate the perturbation of the proper values (50) from the general theory of § 1, caused by including the D-term in (41ʹ). The equation becomes self-adjoint if we multiply by ηn+1. The density function ρ(x) of the general theory thus becomes ηn . As perturbation function r (x ) appears
(51)
(We formally put the perturbation parameter λ = 1; if we desired, we could identify D or F with it.) Now formula (7ʹ) gives, for the perturbation of the kth proper value,
(52)
For the integral in the denominator, which merely provides for the normalisation, formula (115) of the appendix gives the value
(53)
while the integral in the numerator is evaluated in the same place, as
(54)
Consequently
(55)
The condition for the kth perturbed proper value of equation (41ʹ) and therefore, naturally, also for the kth discrete proper value of the original equation (41) runs therefore
(56)
(εk is retained meantime for brevity).
This result is applied twice, namely, to the last two equations of (38) by substituting the two systems (42) of values of the constants A, B, C, D; and it is to be observed that n is the same number in the two cases, while the two k-values are to be distinguished by the suffixes 1 and 2, as above. First we have
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