38 



Supplement to '' Nature,'' July 7, 1923 



states demands another quantum number which we 

 shall call the subordinate quantum number. 



A survey of the motion in the stationary states 

 thus fixed is given in the diagram (Fig. 5), which 

 reproduces the relative size and form of the electron 

 orbits. Each orbit is designated by a symbol n^, 

 where n is the |)rincipal quantum numl)er and k the 

 subordinate quantum number. All orbits with the 

 same principal quantum number have, to a first 

 approximation, the same major axis, while orbits 

 uith the same value of k have the same parameter, 

 / I me value for the shortest chord through the 



l(>*u... Mnce the energy values for different states 

 with the same value of n but different values of k 

 differ a little from each other, we get for each hydrogen 

 line corresponding to definite values of n' and n" \x\. 

 the Balmer formula a number of different transition 

 processes, for which the frequencies of the emitted 

 radiation as calculated by the second postulate are 



Fig. 5. 



not exactly the same. As Sommerfeld was able to 

 show, the components this gives for each hydrogen 

 line agree with the observations on the fine structure 

 of hydrogen lines to within the limits of experimental 

 error. In the figure the arrows designate the processes 

 that give rise to the components of the red and green 

 lines in the hydrogen spectrum, the frequencies of 

 which are obtained by putting «" = 2 and n' = 3 or 4 

 respectively in the Balmer formula. 



In considering the figure it must not be forgotten 

 that the description of the orbit is there incomplete, 

 in so much as with the scale used the slow precession 

 does not show at all. In fact, this precession is so 

 slow that even for the orbits that rotate most rapidly 

 the electron performs about 40,000 revolutions before 

 the perihelion has gone round once. Nevertheless, 

 it is this precession alone that is responsible for the 

 multiplicity of the stationary states characterised by 

 the subordinate quantum number. If, for example, 

 the hydrogen atom is subjected to a small disturbing 

 force which perturbs the regular precession, the 

 electron orbit in the stationary states will have a 

 form altogether different from that given in the figure. 

 This implies that the fine structure will change its 



character completely, but the hydrogen spectrum will 

 continue to consist of lines that are given to a close 

 approximation by the Balmer formula, due to th< 

 fact that the approximately periodic character of th* 

 motion will be retained. Only when the disturbing 

 forces become so large that even during a singl« 

 revolution of the electron the orbit is appreciabl\ 

 disturbed, will the spectrum undergo essential changes. 

 The statement often advanced that the introduction of 

 two quantum numbers should be a necessar>- condition 

 for the explanation of the Balmer formula must there- 

 fore be considered as a misconception of the theory. 



Sommerfeld's theory has proved itself able to 

 account not only for the fine structure of the hydrogen 

 lines, but also for that of the lines in the helium spark 

 spectrum. Owing to the greater velocity of the 

 electron, the intervals between the components into 

 which a line is split up are here much greater and can 

 be measured with much greater accuracy. The theory 

 was also able to account for certain 

 features in the .fine structure of 

 X-ray spectra, where we meet fre- 

 quency differences that may even 

 reach a value more than a million 

 times as great as those of the fre- 

 quency differences for the com- 

 ponents of the hydrogen lines. 



Shortly after this result had been 

 attained, Schwarzschild and Epstein 

 k/)f/ (19^^) simultaneously succeeded, by 

 means of similar considerations, in 

 accounting for the characteristic 

 changes that the hydrogen lines 

 undergo in an electric field, which 

 had been discovered by Stark in 

 the year 1914. Next, an ex 

 planation of the essential featun 

 of the Zeeman effect for the hydro- 

 gen lines was worked out at the 

 same time by Sommerfeld and Debye 

 (191 7). In this instance the applica- 

 tion of the Postulates involved the consequence 

 that only certain orientations of the atom relative 

 to the magnetic field were allowable, and this character- 

 istic consequence of the quantum theor}^ has quite 

 recently received a most direct confirmation in the 

 beautiful researches of Stern and Gerlach on the 

 deflexion of swiftly-moving silver atoms in a non- 

 homogenous magnetic field. 



& 



The Correspondence Principle. 

 While Ihis development of the theor}- of spectra 

 was based on the working out of formal methods 

 for the fixation of stationary' states, the present 

 lecturer succeeded shortly afterwards in throwing 

 light on the theor\- from a new view-point, by pursuing 

 further the characteristic connexion between the 

 quantum theory and classical electrodynamics already 

 traced out in the hydrogen spectrum. In connexion 

 with the important work of Ehrenfest and Einstein 

 these efforts led to the formulation of the so-called 

 correspondence principle, according to which the 

 occurrence of transitions between the stationary states 

 accompanied by emission of radiation is traced back 

 to the harmonic components into which the motion 



