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Perturbation methods in quantum mechanics Pearson, Hans Lennart

Abstract

The solutions of the radial part of the Schrödinger equation for the hydrogen atom, which may be written (in atomic units) as [-1/r² d/dr r² d/dr + ℓ(ℓ + 1)/ r² - 2/r] Ψ(r) = EΨ(r) are well known in the standard case when the boundary conditions require that the wave function should vanish for infinite r . The eigenfunctions in this case are expressible in terms of Laguerre polynomials and the eigenvalues of the energy are E[subscript n] = - 1/n² ( n = 1, 2 ...) The problem of determining the eigenvalues when the boundary conditions require that Ψ should vanish for a finite r , say r₀ , is not as amenable to solution, and it is only recently that several methods have been suggested for dealing with this case. The method to be discussed here is due to Michels, de Boer, and Bijl. De Boer, considering the ground state alone, succeeds through the use of a perturbation method in finding the change in the eigenvalues for different r₀ . In so doing, he makes an approximation, which a priori is not justified. In the present thesis, it is shown both qualitatively and quantitatively that the approximation is justified for the values of r₀ used. The logical extension of the method to states other than the ground state is made for two particular cases, and from the results of these two investigations, conclusions are drawn regarding the general applicability of de Boer's method.

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