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Nobel prize to E. Schrödinger awarded in 1933. Co-winner P. A. M. Dirac "for the discovery of new productive forms of atomic theory'' |  |
SCHRÖDINGER 1926D
Schrödinger, E.;
Quantizierung als Eigenwertproblem (Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien) / Quantization as a Problem of Proper Values. Part III: Perturbation Theory, with Applications to the Stark Effect of Balmer lines
Annalen der Physik. Leipzig 80 (1926) 437;
Reprinted in
Abhandlungen zur Wellenmechanik, Schrödinger, (1927d) 85.
Gesammelte Abhandlungen / Collected Works E.Schrödinger (1984c) 166.
(translation into English) Collected Papers on Wave Mechanics by E. Schrödinger, Glazgow (1928) 62.
Introduction
As has already been mentioned at the end of the preceding paper, ( last two paragraphs of Part II.) 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, (Courant-Hilbert, chap. VI, S S 2, 4, p. 337) 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 investigating (Courant-Hilbert, chap. V. S 5, 2, p.241) 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 generalization 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 generalizing to the widest possible extent than on bringing forward the very simple rudiments in the clearest possible manner. From the latter, any desired generalization 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 solved (P. S. Epstein, Ann. d. Phys. 50 (1916) 489) 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. (N. Bohr, Kopenhagener Akademie (8)IV., 1, 2 (1918) 69 et seq.) The first method will be utilized 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. (A. Sommerfeld, Atombau, 4th ed. 772). In the second method, it is found that in the case of the Stark effect
an exact separation coordinate 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 generalization.
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. (H. A. Kramers, Kopenhagener Akademie (8), III., 3 (1919) 287).
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, ( Physica, 1925; Die Naturwissenschaften, 1926; Nature, 20th Feb. 1926;
cf. also L. H. Thomas, Nature, 10th April, 1926.) 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 Slater (Proc. Amer. Nat. Acad. 11 (1925) 732) and by Sommerfeld and Unsöld (Ztschr.
f. Phys. 36 (1926) 259) 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, 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.
Related references
More (earlier) information appears in
E. Schrödinger, Annalen der Physik. Leipzig 79 (1926) 489;
E. Schrödinger, Naturw. 14 (1926) 664;
See also
W. Pauli, Z. Phys. 36 (1926) 336;
N. Bohr, Naturw. 1 (1926) 1;
A. Sommerfeld and A. Unsoeld, Z. Phys. 36 (1926) 259;
J. C. Slater, Proc. Amer. Nat. Acad. 11 (1925) 732;
L. H. Thomas, Nature, 10th Apr., (1926);
A. Sommerfeld, Atombau und Spectrallinien 4th ed. 775;
Courant-Gilbert, Methods of Mathematical Physics I. Berlin, Springer (1924) 26;
Analyse data from
J. Stark, Annalen der Physik. Leipzig 43 (1914) 1001;
J. Stark, Annalen der Physik. Leipzig 48 (1915) 193;
R. Ladenburg, Z. Phys. 38 (1926) 249;
R. Ladenburg and F. Reiche, Naturw. 11 (1923) 584;
M. Born and P. Jordan, Z. Phys. 34 (1925) 886;
P. S. Epstein, Annalen der Physik. Leipzig 50 (1916) 489;
W. Heisenberg, Z. Phys. 33 (1925) 879;
G. E. Uhlenbeck and S. Goudsmit, Naturw. 13 (1925) 953;
H. A. Kramers, Kopenhagener Akademie (8) III (1919) 287;
N. Bohr, Kopenhagener Akademie (8) IV. (1918) 69;
M. Born and P. Jordan, Z. Phys. 34 (1925) 867;
