294 



10 



of coordinates, given by Jacobi^), that the Cartesian coordinates are functions of 

 the elliptical coordinates of the type (16). In the applications of the quantum theory 

 hitherto made, separation of variables is always obtained in one or other set of 

 elliptical coordinates^), and, due to the special form of the expression for the kinetic 

 energy in mechanics, it seems highly questionable if, for a mechanical system con- 

 sisting of particles moving under the influence of conservative forces, it is possible 

 to obtain separation of variables in other kinds of coordinates. 



§ 2. Hydrogen atom undisturbed by external influences. 



In this chapter we shall apply the above analysis to the problem of the motion 

 of an electron of mass m and charge —e rotating round a positive nucleus of infi- 

 nite mass and of charge A^e, which attracts the electron according to Coulomb's 

 law, assuming that the motion is governed by relativistic mechanics. As well known 

 this system represents the model of a hydrogen atom where the mass of the nucleus 

 is regarded as infinite. If the laws of Newtonian mechanics were applied, the electron 

 would perform a periodic Keplerian motion, but as soon as the modifications in 

 the laws of mechanics, claimed by the theory of relativity, are taken into 

 account the motion will no more be simply periodic. The orbit of the electron 

 will, however, still be plane and may be described as a closed periodic orbit on 

 which a uniform rotation round the nucleus is superposed. Moreover, assuming 

 that the velocity v of the electron is small compared to the velocity c of light, the 

 closed orbit in question will differ from a Keplerian orbit only by small quantities 

 of the same order of magnitude as "'/c-, while also the ratio of the frequency o of 

 the superposed rotation to the frequency of revolution of the electron in the closed 

 orbit will be of the same order as "'/c-. 



From these simple properties of the motion it would be possible, quite inde- 

 pendently of the theory of separation of variables, at once to derive trigonometric 

 series expressing the displacement of the electron in different directions as a function 

 of the time with neglect of small quantities of the order «^-/c-. In fact, the expansions 

 in a trigonometric series for the Cartesian coordinates f and of a point describing 

 a closed Keplerian ellipse are well known in celestial mechanics, and from these 

 expansions are easily obtained the expressions for the Cartesian coordinates x and 

 y in a fixed system of coordinates, relative to which the system rotates uniformly 

 with the frequency 0. An example of a procedure of this kind will be given at 

 the end of this chapter, where the influence of a magnetic field on the motion of 

 the electron in the hydrogen atom will be treated. For the present, however, we 

 will for the sake of illustration treat the problem by means of the general method 



') Jacobi, Vorl. über Dynamik, p. 202. 



Rectangular coordinates, polar coordinates and parabolic coordinates may all be regarded as 

 special cases of elliptical coordinates. 



