Selection Rules for the Kepler Problem

As was shown in 6 (p. 130), the radiation associated with a quantum jump is given essentially by the matrix element of the co-ordinate, which is related to the wave-mechanical mean value of the electrical dipole moment in the way indicated in the text  

To do this we must investigate the proper functions for the various states. We deduced these in Appendix XVIII (p. 298), and showed that they may be written in the form 

We recall that the functions Yj are obtained by removing the factor faom a homogeneous polynomial of the Z-th degree, Ui(x, y, z), which satisfies Laplace s equation AZ7j = 0. If we introduce this form of the wave functions into the matrix element above, the integral is split up into two parts, an integral over the elementary solid angle d(jo = sm 9 ddd(f , which is of the form 

We now assert that is always zero unless the selection rule = Z + 1 is satisfied. To prove this some preliminaries are required. In the first place, it is easy to see that the integral 

If we substitute this in the int( gral Jit follows from the relationjiist proved that the integral vanishes identically except when Z = Z j- 1. 

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