688 BELL SYSTEM TECHNICAL JOURNAL 



The results are, in fact, just like those obtained with the Bohr- 

 Sommerfeld atom-model; and this is somewhat embarrassing. For, 

 in order to perfect the Bohr-Sommerfeld model and establish a com- 

 plete analogy between (for instance) the sodium spectrum on the one 

 hand and the fine-structure of the hydrogen lines on the other hand, 

 it was necessary to introduce a new feature — the "spinning electron." 

 Something of the sort must evidently be done again — the "spinning 

 electron" must be imported into the undulatory mechanics; but the 

 exact way to do it seems as yet to elude the virtuosi of mathematical 

 physics. ^^ 



In one case — when the perturbing force is an Impressed electric 

 field — the results obtained by the method of Bohr and Sommerfeld 

 and those obtained by the method of Schroedinger agree to first 

 approximation with each other and with the data of experience, 

 without the introduction of a "spinning electron." As this case of 

 the "Stark Effect" furnishes a convenient transition to the last section 

 of the article, I will quote the results. ^^ 



The Stark Effect 



Imagine a hydrogen atom, upon which an electric field F parallel 

 to some arbitrary direction which we call the ^-direction is acting. 

 Owing to this field, the electron at the point x, y, z and the nucleus 

 at the origin (we are still using the concept of the nucleus and the 

 electron!) possess a potential energy composed of the "intrinsic" 

 term — e'^jr and the "perturbation" -f eFz. The wave-equation 

 takes the form: 



V2^-f ^/E+f-'-^Fz) = 0. (183) 



Paraboloidal coordinates are indicated for this problem. Instead 

 of the planes, double-cones and spheres of the polar coordinate-system 

 which we earlier used, it is desirable to employ planes and two families 

 of paraboloids of rotation ; the planes intersect one another along the 

 line through the nucleus parallel to the field (hitherto called the z-axis), 

 and the two families of paraboloids have their common foci at the 

 nucleus and their noses pointing opposite ways along that axis. The 

 transformation is made by the equations: 



X = V^Tjcos (f, y = Vlijsin (/?, s = |(^ — r?) (184) 



^^ Unless the problem has been solved by C. F. Richter (cf. preliminary note in 

 Proc. Nat. Acad. Sci., 13, pp. 476-i79; 1927). 



19 Schroedinger, Ann. d. Fhys., 80, pp. 457-464 (1926); P. S. Epstein, P/jj^. Rev. (2), 

 28, pp. 695-710 (1926). 



