Introduction

In an attempt to develop certain outlines of a theory of line-spectra based on a suitable application of the fundamental ideas introduced by Planck in his theory of temperatureradiation to the theory of the nucleus atom of Sir Ernest Rutherford, the writer has shown that it is possible in this way to obtain a simple interpretation of some of the main laws governing the line-spectra of the elements, and especially to obtain a deduction of the well known Balmer formula for the hydrogen spectrum1)

The theory in the form given allowed of a detailed discussion only in the case of periodic systems, and obviously was not able to account in detail for the characteristic difference between the hydrogen spectrum and the spectra of other elements, or for the characteristic effects on the hydrogen spectrum of external electric and magnetic fields.

Recently, however, a way out of this difficulty has been opened by Sommerfeld2 ) who, by introducing a suitable generalisation of the theory to a simple type of non-periodic motions and by taking the small variation of the mass of the electron with its velocity into account, obtained an explanation of the fine-structure of the hydrogen lines which was found to be in brilliant conformity with the measurements. Already in his first paper on

) N. Bohr, Phil. Mag., XXVI, pp. 1, 476, 857 (1913), XXVII, p. 506 (1914), XXIX. p. 332 (1915), XXX. p. 394 (1915). 2 ) A. Sommerfeld, Ber. Akad. M¨unchen, 1915, pp. 425, 459, 1916, p. 131. 1917. p. 83. Ann. de Phys., LI. p. 1 (1916). 2 this subject, Sommerfeld pointed out that his theory evidently offered a clue to the interpretation of the more intricate structure of the spectra of other elements. Briefly afterwards Epstein1 ) and Schwarzschild, 2 ) independent of each other, by adapting Sommerfeld’s ideas to the treatment of a more extended class of non-periodic systems obtained a detailed explanation of the characteristic effect of an electric field on the hydrogen spectrum discovered by Stark.

Subsequently Sommerfeld3 ) himself and Debye4 ) have on the same lines indicated an interpretation of the effect of a magnetic field on the hydrogen spectrum which, although no complete explanation of the observations was obtained, undoubtedly represents an important step towards a detailed understanding of this phenomenon.

In spite of the great progress involved in these investigations many difficulties of fundamental nature remained unsolved, not only as regards the limited applicability of the methods used in calculating the frequencies of the spectrum of a given system, but especially as regards the question of the polarisation and intensity of the emitted spectral lines.

These difficulties are intimately connected with the radical departure from the ordinary ideas of mechanics and electro1 ) P. Epstein, Phys. Zeitschr. XVII, p. 148 (1916), Ann. d. Phys. L, p. 489. LI. p. 168 (1916). 2 ) K. Schwarzschild, Ber. Akad. Berlin, 1916, p. 548. 3 ) A. Sommerfeld, Phys. Zeitschr. XVII, p. 491 (1916). 4 ) P. Debye, Nachr. K. Ges. d. Wiss. G¨ottingen, 1916, Phys. Zeitschr. XVII, p. 507 (1916). 3 dynamics involved in the main principles of the quantum theory, and with the fact that it has not been possible hitherto to replace these ideas by others forming an equally consistent and developed structure. Also in this respect, however, great progress has recently been obtained by the work of Einstein1 ) and Ehrenfest. 2 ) On this state of the theory it might therefore be of interest to make an attempt to discuss the different applications from a uniform point of view, and especially to consider the underlying assumptions in their relations to ordinary mechanics and electrodynamics. Such an attempt has been made in the present paper, and it will be shown that it seems possible to throw some light on the outstanding difficulties by trying to trace the analogy between the quantum theory and the ordinary theory of radiation as closely as possible. The paper is divided into four parts. Part I contains a brief discussion of the general principles of the theory and deals with the application of the general theory to periodic systems of one degree of freedom and to the class of non-periodic systems referred to above. Part II contains a detailed discussion of the theory of the hydrogen spectrum in order to illustrate the general 1 ) A. Einstein, Verh. d. D. phys. Ges. XVIII, p. 318 (1916), Phys. Zeitschr. XVIII, p. 121 (1917). 2 ) P. Ehrenfest, Proc. Acad. Amsterdam, XVI. p. 591 (1914), Phys. Zeitschr. XV. p. 657 (1914), Ann. d. Phys. LI. p. 327 (1916), Phil. Mag. XXXIII. p. 500 (1917). 4 considerations. Part III contains a discussion of the questions arising in connection with the explanation of the spectra of other elements. Part IV contains a general discussion of the theory of the constitution of atoms and molecules based on the application of the quantum theory to the nucleus atom. Copenhagen, November 1917.