Application of Perturbation Theory to Hydrogenic Systems

We can set the charge Z = 1 since it can easily be put back into the final result if desired. Thus, 

The matrix elements of W can be obtained from those of r and z given in Tables I and II (Appendix C) by simple matrix multiplication. They are defined by 

A special property of the perturbation series for the energy is that all odd-order corrections are zero. Thus, using Eq. (274) and the symmetric perturbation formula Eq. (275), the first few orders are 

For the first form of the energy expansion the coefficients up to 20th order are given in Table V, Appendix C, and the coefficients using the second form of the expansion have been given to order 60 by Privman (1980). 

The study of the hydrogen atom in a uniform magnetic field (HAMF) is considerably more complex than the LoSurdo-Stark effect (see Cizek and Vrscay, 1982, and references therein). The Hamiltonian is not separable and reducible to a one-dimensional problem. For a field along the z axis the problem is inherently two-dimensional. Thus, the methods mentioned above which rely on the one-dimensional aspect of the LoSurdo-Stark effect and its separability in parabolic coordinates are special and not directly extendable to the Zeeman effect. 

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