;uo 



26 



§ 4. Hydrogen atom under the influence of a weak homogeneous electric 



field of force. 



In this section we shall consider the mechanical problem of the motion of an 

 electron which is subject to the attraction of a positive nucleus of infinite mass as 

 well as to the influence of a weak homogeneous electric field of force, and which moves 

 according to the laws of relativistic mechanics. The general case of this problem 

 in which the intensity of the electric force may have any value so that the devia- 

 tions of the motion of the electron from a simple Keplerian motion, due to the 

 influence of the relativity modifications in the laws of mechanics, must be considered 

 as being of the same order of magnitude as those due to the electric field will be 

 treated in a later paper which deals with the general problem of the effect of an 

 electric field on the fine structure of the hydrogen lines. In this section we vvill 

 only consider the special case in which the electric field is so weak that its influ- 

 ence on the motion of the electron is small compared with the influence which is 

 due to the relativity modifications. 



Let the nucleus be situated at the origin of a system of rectangular Cartesian 

 coordinates x, y, z, the z-axis of which is taken parallel to the direction of the 

 electric force. The mass and charge of the electron will again be denoted by m 

 and — e respectively and the charge of the nucleus by Ne, while the intensity of the 

 electric field will be denoted by F. Let further / be a small quantity of the same 

 order of magnitude as the square of the ratio between the velocity of the electron 

 and the velocity of light, and f a small quantity of the same order of magnitude 

 as the ratio between eF and the forces which the nucleus exerts on the electron. We 

 shall according to the above assume that f is small compared to Å, and it will 

 be our purpose to solve the equations of motion retaining only small quantities of 

 the same order as / and fU, and neglecting all quantities of higher order of magni- 

 tude such as f, etc. in the expressions for the coordinates x, y, z of the electron 

 as functions of the time. 



Let us introduce polar coordinates r, t^, ^, which in the well known way are 

 connected with x, y, z by the formulae 



z = rcosâ, X -\- i y = rs'möe^v. 



The velocity v of the electron will then be given by u-' = (dridtf + r'H'^'^ldty 



fiin^ H{dyjdtf. Introducing the notation = (1 - v-lc^)-'lt where c is the velocity 

 of light, the canonically conjugated momenta of r, ë, <p are given by 



Pr - r ^ /V/ = r , p<p = m rr- sm- ä -J , 



and the equations of motion will be of the canonical form (2) where the energy E, 

 expressed as a function of the coordinates and momenta, will be given by^) 



') Compare for instance A. Sommerfeld, Phys. Zeitschr. XVII, p. 506 (1916). See also § 2, page 11. 



