﻿868 Mr. J. H. Van Vleck on the normal Helium Atom 



the two negative electrons, so that the total potential energy 

 V of the system is 



v= ~ 7ww + m (c/ equation (2)) - 



For a = there is no mutual action between negative electrons, 

 and then each electron describes a circle or ellipse character- 

 istic of a central force obeying the inverse square law. 

 Since the motion in this undisturbed system is periodic, the 

 action integral must be an integral multiple of h, so that we 

 have 



T 



2iTdt = ( ni + n 2 )h, (24) 



o 

 where t is the period. Also we shall assume that the 

 resultant angular momentum about the axis of symmetry is 



an integral multiple of ~ — , which apparently is demanded by 



the conservation of angular momentum, and which is required 

 if Sommerfeld's quantizing of space in polar coordinates is 

 accepted. Now let the parameter a be increased from the 

 fictitious value zero to the actual value unity. Since there 

 are no forces operative which have a moment about the axis 

 of symmetry, the resultant angular momentum retains its 



original value —-, and the axial symmetry is preserved. 



LIT 



Also if we assume that the motion always remains condition- 

 ally periodic as the perturbing field is thus gradually 

 increased, then, using an equation given by Ehrenfest *, it 

 is readily shown that the average kinetic energy has the 

 value demanded by (22). 



Bohr's Quantum Conditions, 



By quantizing the perturbations in a manner analogous to 

 that of the Sommerfeld conditions for conditionally periodic 

 motions, Bohr has devised a general method for determining 

 the " allowed " motions whenever the perturbing potential 

 has axial symmetry f, although his treatment is intended 

 primarily for cases where the departures from the undisturbed 

 orbits are so small that only first powers in the perturbations 

 need be considered. These quantum conditions demand that 



# Ann. d. Phys. vol. li. p. 348, equation (m). 



t ' The Quantum Theory of Line Spectra,' pp. 53-6. 



