CONTEMPORARY ADVANCES IN PHYSICS 087 



tions" such as those above (which are not necessarily the only self- 

 consistent nor the best ones) make it possible to translate orbits of 

 the orbit-model into modes of vibration of the wave-model, and vice 

 versa; and to devise definitions for these three kinds of quantum- 

 numbers from the qualities of the A'ibrations themselves. 



Fertui'bations 



Inasmuch as the wa\'e- mechanics indicates n different Eigetifunk- 

 tionen with « different collections of nodal spheres (not to speak of 

 the still more greatly varied possibilities of nodal planes and double- 

 cones) for the Stationary State having the Eigenwert and energy-value 

 En, one may well ask whether there is any chance of distinguishing 

 which of these, or which linear combination of these (for the differ- 

 ential equation will permit any) is actually adopted by a hydrogen 

 atom. 



Translating into the language of the Bohr-Sommerfeld atom-model, 

 we find the question in this form: is there any way of distinguishing 

 which of the n permitted elliptical electron-orbits is actually adopted? 



When the question was asked in this form, it was answered by 

 pointing out that if the force exerted upon the electron were not the 

 pure inverse-square force ascribed to the nucleus, but the sum of this 

 and a perturbing force, the energy-values of the w permitted ellipses 

 would cease to coincide exactly. If for instance the atom under 

 examination were composed of a nucleus of charge 11^, a group of ten 

 electrons very close to it and an "outer" electron relatively far out 

 (the conventional model for a sodium atom in certain states) ; then 

 the group of ten inner electrons would act upon the outer one with a 

 perturbing force, and the n permitted ellipses of the outer one would 

 be endowed with distinct energy-values — the single Stationary State 

 of the outer electron would be dissolved into n distinguishable states. 

 Even in the hydrogen atom, the dependence of the mass of the electron 

 upon its speed should separate the energy-values of the various ellipses 

 which but for this fact would share a common energy En, and produce 

 the fine-structure of the hydrogen lines. 



The very same thing occurs in wave-mechanics; and from the effect 

 of a perturbing force, allowance for which is made in the potential- 

 energy-function introduced into the wave-equation, we may expect 

 to be able to distinguish the different modes of vibration attributed to 

 a single Eigenwert and a single Stationary State of the unperturbed 

 hydrogen atom." 



^Mn the language of the mathematicians, the perturbing forces remove the de- 

 generacy of the problem; some kinds of perturbation remove it completely, others 

 in part. 



