Application of Rayleigh—Schrödinger Perturbation Theory to the Hydrogen Atom. II

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Abstract
Various unsuccessful attempts to treat the hydrogen atom problem by perturbation theory, starting from a free particle as the unperturbed state, are examined. Calculations in closed form are presented, confirming the incorrect results but throwing light on the reasons for the failure, which occurs in spite of the fact that convergent power series expansions in the perturbation parameter exist, and of course are known explicitly. The failure is shown to be essentially due to nonuniform convergence at the point where the boundary conditions are applied, and furthermore this failure occurs for a wide class of potentials besides the Coulomb. Previous methods of avoiding this difficulty for the Coulomb potential are shown not to be of general validity.