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Physics Letters A

Path integral treatment of Coulomb potential by exact summation of a perturbation series

K.V. Bhagwat, S. V. Lawande

Published Mar 1, 1989• 5 citations
Publisher
DOI
10.1016/0375-9601(89)90038-8

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A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ∫d3rβK(rβ,0), where K(rβ,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a ``modified'' perturbation expansion (W is expanded as a power series in β with β = 1/kT). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.
The path integral treatment of the step potential based on exact summation of the Feynman perturbation series is presented in its original form. With a two-sided Laplace transform on the end point, we have established a recurrence formula between three successive terms of the perturbation series which leads us to sum up exactly all the terms and deduce the Green function of the problem. The Wiener–Hopf method and Hilbert problems are successfully used for solving our problem with the boundary conditions.