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arXiv:quant-ph/9901052 · Journal of Physics A Mathematical and General

Green's function for the relativistic Coulomb system via sum over perturbation series

De‐Hone Lin

Published Sep 18, 1998• 14 citations
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Abstract

We evaluate the Green's function of the D -dimensional relativistic Coulomb system via sum over perturbation series which is obtained by expanding the exponential containing the potential term in the path integral into a power series. The energy spectra and wavefunctions are extracted from the resulting amplitude.
DOI
10.1088/0305-4470/31/37/015

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A method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov–Bohm–Coulomb system is obtained by the method. Different from the earlier treatment based on the space–time transformation and infinite multiple-valued trasformation of Kustaanheimo–Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green’s function and an explicit path integral is avoided.
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.