where for brevity
(68)
The dash on the Laplacian operator is merely to signify that in it the letter η is to be written for the radius vector.
In equation (67) we conceive l to be the proper value, and the term in g to be the perturbing term. The fact that the perturbing term contains the proper value need not trouble us in the first approximation. If we neglect the perturbing term, the equation has as proper values the natural numbers
(69)
and no others. (The extended spectrum is again cut out by the artifice (66), which would be valuable for closer approximations.) The allied proper functions (not yet normalised) are
(70)
Here signifies the mth “associated” Legendre function of the nth order, and is the (2n + 1)th derivative of the (n + l)th Laguerre polynomial.fi So we must haven < l,
otherwise would vanish, because the number of differentiations would be greater than the degree. With reference to this, the numbering of the spherical surface harmonics shows that l is an lfj-fold proper value of the unperturbed equation. We now investigate the splitting up of a definite value of l, supposed fixed in what follows, due to the addition of the perturbing term.
To do this we have, in the first place, to normalise our proper functions (70), according to § 2. From an uninteresting calculation, which is easily performed with the aid of the formulae in the appendix,fj we get as the normalising factor
(71)
if m ≠ 0, but, for m = 0, times this value. Secondly, we have to calculate the symmetrical matrix of constants εim, according to (22). The r there is to be identifiedfj with our perturbing function −gη3 cos θ sin θ, and the proper functions, there called uki, are to be identified with our functions (70). The fixed suffix k, which characterises the proper value, corresponds to the first suffix l of ψ lnm, and the other suffix i of uki corresponds now to the pair of suffixes n, m in ψlnm. The matrix (22) of constants forms in our case a square of l2 rows and l2 columns. The quadratures are easily carried out by the formulae of the appendix and yield the following results. Only those elements of the matrix are different from zero, for which the two proper functions ψlnm , ψlnʹm’, to be combined, satisfy the following conditions simultaneously:1. The upper indices of the “associated Legendre functions” must agree, i. e. m = m’.
2. The orders of the two Legendre functions must differ exactly by unity, i.e. |n − n’| = 1.
3. To each trio of indices lnm, if m ≠ 0, there belong, according to (70), two Legendre functions, and thus also two proper functions ψlnm, which only differ from each other in that one contains a factor cos mφ and the other sin mφ. The third condition reads: we may only combine sine with sine, or cosine with cosine, and not sine with cosine.
The remaining non-vanishing elements of the desired matrix would have to be characterised from the beginning by two index-pairs (n, m) and (n + 1, m). (We renounce any idea of showing the fixed index l explicitly.) Since the matrix is symmetrical, one index pair (n, m) is sufficient, if we stipulate that the first index, i.e. n, shall mean the greater of the two orders n, n’, in every case.
Then the calculation gives
(72)
We have now to form the determinant (22) out of these elements. It is advantageous to arrange its rows as well as its columns on the following principle. (To fix our ideas, let us speak of the columns, and therefore of the index-pair characterising the first of the two Legendre functions.) Thus: first come all terms with m = 0, then all with m = 1, then all with m = 2, etc., and finally, all terms with m = l − 1, which last is the greatest value that m (like n) can take. Inside each of these groups, let us arrange the terms thus: first, all terms with cos mφ, and then all with sin mφ. Within these “half groups” let us arrange them in order of increasing n, which runs through the values m, m + 1, m + 2 ... l − 1, i.e. (l − m) values in all.
If we carry this out, we find that the non-vanishing elements (72) are exclusively confined to the two secondary diagonals, which lie immediately alongside the principal diagonal. On the latter are the proper value perturbations which are to be found, but taken negatively, while everywhere else are zeros. Further, the two secondary diagonals are interrupted by zeros at those places, where they break through the boundaries between the so-called “half-groups”, in very convenient fashion. Hence the whole determinant breaks up into a product of just so many smaller determinants as there are “half-groups” present, viz. (2l − 1). It will be sufficient if we consider one of them. We write it here, denoting the desired perturbation of the proper value by ε (without suffix):
(73)
If we divide each term here by the common factor 6lg of the ∈nm’s (cf. (72)), and for the moment regard as the unknown
(74)
the above equation of the (l − m)th degree has the roots
(75)
where the series stops with ± 1 or 0 (inclusive) according as the degree l − m is even or odd. The proof of this is unfortunately not to be found in the appendix, as I have not been successful in obtaining it.
If we form the series (75) for each of the values m = 0, 1, 2 . . . (l − 1), then we have in the numbers
(76)
the complete set of perturbations of the principal quantum number l. In order to find the perturbed proper values E (term-levels) of the equation (32), we have only to substitute (76) in
(77)
taking into account the signification of the abbreviations g (see (68)) and a0 (see (63)).
After reducing this gives
(78)
Comparison with (62) shows that k* is the difference k2 − k1 of the parabolic quantum numbers. From (75), bearing in mind the range of values of m referred to above, we see that k* may also take the same values as the difference just mentioned, viz. 0, 1, 2 . . . (l − 1). Also, if we take the trouble to work it out, we will find for the multiplicity, in which k* and the difference k2 − k1 appear, the same value, viz. l—|k*|.
l − We |k*|. have thus obtained the proper value perturbations of the first order also from the general theory. The next step would be the solution of the system (21’) of linear equations of the general theory for the κ-quantities. These would then yield, according to (18) (provisionally putting λ = 0), the perturbed proper functions of zero order; this is nothing more than a representation of the proper functions (64) as linear forms of the proper functions (70). In our case the solution of (21’) would naturally be anything but unique, on account of the considerable multiplicity of the roots ε. The solution is made much simpler if we notice that the equations break up into just as many groups, viz. (2l − 1), or, retaining the former expression, half-groups, with completely separated variables, as the determinant investigated above contains factors like (73); and if we further notice that it is allowable, after we have chosen a definite ∈-value, to regard only the variables κ of a single half-group as different from zero, of that half-group, in fact, for which the determinant (73) vanishes for the chosen ∈-value. The definition of this half-group of variables is then unique.
But our object, viz. to illustrate the general method of § 2 by an example, has been sufficiently attained. Since the continuation of the calculation is of no special physical interest, I have not troubled to bring the determinantal quotients, which we immediately obtain for the coefficients κ, into a clearer form, or to work out the transformation to principal axes in any other way.
On the whole, we must admit that in the present case the method of secular perturbations (§ 5) is considerably more troublesome than the direct application of a system of separation (§ 3). I believe that this may ako be true in other cases. In ordinary mechanics it is, as we know, usually quite the reverse.
III. MATHEMATICAL APPENDIX
Prefatory Note:—It is not intended to supply in uninterrupted detail all the calculations omitted from the text. Without that, the present paper has already become too long. In general, only those methods of calculation will be briefly described which a
nother might utilise with advantage in similar work, if something better does not occur to him—as it may easily do.
§ 1. THE GENERALISED LAGUERRE POLYNOMIALS AND ORTHOGONAL FUNCTIONS
The kth Laguerre polynomial Lk (x) satisfies the differential equationfk
(101)
If we first replace k by n + k, and then differentiate n times, we find that the nth derivative of the (n + k)th Laguerre polynomial, which we will always denote by , satisfies the equation
(102)
Moreover, by an easy transformation, we find that for the following equation holds,
...(103)
This found an application in equation (41ʹ) of § 3. The allied generalised Laguerre orthogonal functions are
(104)
Their equation, it may be remarked in passing, is
(105)
Let us turn to equation (103), and consider there that n is a fixed (real) integer, and k is the proper value parameter. Then, according to what has been said, in the domain x ≥ 0, at any rate, the equation has the proper functions,
(106)
belonging to the proper values,
(107)
In the text it is maintained that it has no further values, and, above all, that it possesses no continuous spectrum. This seems paradoxical, for the equation
(108)
into which (103) is transformed by the substitution
(109)
does possess a continuous spectrum, if in it we regard
(110)
as proper value parameter, viz. all positive values of E are proper values (cf. Part I., analysis of equation (7)). The reason why no proper values k of (103) can correspond to these positive E-values is that by (110) the k-values in question would be complex, and this is impossible, according to general theorems.fl Each real proper value of (103), by (110), gives rise to a negative proper value of (108). Moreover, we know (cf. Part I.) that (108) possesses absolutely no negative proper values other than those that arise, as in (110), from the series (107). There thus remains only the one possibility, that in the series (107) certain negative k-values are lacking, which appear on solving (110) for k, on account of the double-valuedness when extracting the root. But this also is impossible, because the k-values in question turn out to be algebraically less than −n + and thus, from general theorems,fm cannot be proper values of equation (103). The series of values (107) is thus complete. Q.E.D.
The above supplements the proof that the functions (70) are the proper functions of (67) (with the perturbing term suppressed), allied to the proper values (69). We have only to write the solutions of (67) as a product of a function of θ , φ and a function of η. The equation in η can readily be brought to the form of (105), the only difference being that out present n is there always an odd number, namely, the (2n + 1) which is to be found there.
§ 2. DEFINITE INTEGRALS OF PRODUCTS OF TWO LAGUERRE ORTHOGONAL FUNCTIONS
The Laguerre polynomials can all be obtained, in the following manner, as coefficients of the powers of the auxiliary variable t, in the expansion in a series of a so-called “generating function”fn
(111)
If we replace k by n + k and then differentiate n times with respect to x , we obtain the generating function of our generalised polynomials,
(112)
In order to evaluate with its help integrals such as appeared for the first time in the text in expression (52), or, more generally, such as were necessary in § 4 for the calculation of (65), and also in § 5, we proceed as follows. We write (112) over again, providing both the fixed index n and the varying index k with a dash, and replacing the undefined t by s. These two equations are then multiplied together, i.e. left side by left side, and right side by right. Then we multiply further by
(113)
and integrate with respect to x from 0 to ∞. p is to be a positive integer—this being sufficient for our purpose. The integration is practicable by elementary methods on the right-hand side, and we get
(114)
We have now, on the left, the desired integrals like pearls on a string, and we merely detach the one we happen to need by searching on the right for the coefficient of tksk’ . This coefficient is always a simple sum, and, in fact, in the cases occurring in the text, always a finite sum with very few terms (up to three). In general, we have
(115)
The sum stops after the smaller of the two numbers k, kʹ. It often, in actual fact, begins at a positive value of τ, as binomial coefficients, whose lower number is greater than the upper, vanish. For example, in the integral in the denominator of (52), we put p = n = nʹ, and kʹ = k. Then τ can take only the one value k, and we can establish statement (53) of the text. In the integral of the numerator in (52), only p has another value, namely p = n + 2. τ now takes the values k − 2, k − 1, and k , and after an easy reduction we get formula (54) of the text. In the very same way the integrals appearing in § 5 are evaluated by Laguerre polynomials.
We can now, therefore, regard integrals of the type of (115) as known, and we have only to concern ourselves with those occurring in § 4 in the calculation of intensities (cf. expression (65) and functions (64) which have, to be substituted there). In this type, the two Laguerre orthogonal functions, whose product is to be integrated, have not the same argument, but, for example, in our case, have the arguments λ1/la0 and λ1/l’a0, where l and lʹ are the principal quantum numbers of the two levels that we have combined. Let us consider, as typical, the integral
(116)
Now we can proceed in a superficially different way. At first, the former procedure still goes on smoothly; only on the right-hand side of (114) a somewhat more complicated expression appears. In the denominator occurs the power of a quadrinomial instead of that of a binomial, as before. And this makes the matter somewhat confusing, for the right-hand side of (114) becomes five-fold instead of three-fold, and thus the right side of (115) becomes a three-fold instead of a simple sum. I found that the following substitution made things clearer:
(117)
Hence
(118)
After expanding the two polynomials in their Taylor series, which are finite and have similar polynomials as coefficients, we get, using the abbreviations
(119)
the following,
(120)
Thus the calculation of J is reduced to the simpler type of integral (115). In the case of the Balmer lines, the double sum in (120) is comparatively tractable, for one of the two k-values, namely, the one referring to the two-quantum level, never exceeds unity, and thus λ may have two values at most, and, as it turns out, µ four values at most. The circumstance that out of the polynomials referring to the two-quantum level, none but
appear, permits further simplifications. Nevertheless we must calculate out a number of tables, and it is much to be regretted that the figures given in the tables of the text for the intensities do not allow their general construction to be seen. By good fortune the additive relations between the ||- and the ⊥ components hold good, so that we may, with some probability, feel ourselves safe from arithmetical blunders at least.
§ 3. INTEGRALS WITH LEGENDRE FUNCTIONS
There are three simple integral relations between associated Legendre functions, which are necessary for the calculations in § 5. For the convenience of others, I will state them here, because I was not able to discover them in any of the places I searched. We use the customary definition,
(121)
Then the following holds,
(122)
Moreover,
(123)
On the other hand,
(124)
The last two relations decide the “selection” of the determinantal terms on page 408 of the text. They are, moreover, of fundamental importance for the theory of spectra, for it is obvious that the selection principle for the azimuthal quantum number depends on them (and on two others which have sin2 θ in place of cos θ sin θ).
Addition at Proof Correction
The Dreams That Stuff is Made of Page 33