Application of the Rayleigh-Schrödinger Perturbation Theory to the Hydrogen Atom

Abstract

Wigner has calculated the ground-state energy of the hydrogen atom with second-order perturbation theory, the whole electrostatic potential being considered as the perturbation. Though his result is finite, it does not agree with the known energy for hydrogen. A fact implicit in the literature, but not usually appreciated in this connection, is pointed out—that if part of the electrostatic energy is retained in the zero-order problem, a correct result is obtained, even if the part of the energy retained is given a zero magnitude after the calculation is carried out. It is also shown that if the calculation is carried out in an Einstein hypersphere, the entire "electrostatic potential" can be regarded as the perturbation and the correct result is obtained. Perturbation theory leads to the exact energy eigenvalue in all approximations higher than the first, but the expansion of the eigenfunction converges slowly if the perturbation is made large.