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arXiv:quant-ph/9909050 · Journal of Mathematical Physics

Path integral solution by sum over perturbation series

De‐Hone Lin

Published May 1, 2000• 9 citations
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Abstract

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.

DOI
10.1063/1.533266

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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.
Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.
The Green's functions of the two and three-dimensional relativistic Aharonov-Bohm (A-B) systems are given by the path integral approach. In addition the exact radial Green's functions of the spherical A-B quantum billiard system in two and three-dimensional are obtained via the perturbation techanique of δδδ-function.
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.