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Journal of physics. A, Proceedings of the Physical Society. General

Path integration of a three body problem

Dinkar C. Khandekar, S. V. Lawande

Published Jul 1, 1972• 3 citations
Publisher

Abstract

Path integration for a one dimensional three body problem is discussed. This involves essentially the path integration of a Lagrangian in plane polar coordinates with a potential energy term depending on both the radial and angular coordinates.
DOI
10.1088/0305-4470/5/7/001

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The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring-potential, the Coulomb potential with an Aharonov-Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring-potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.
Using a scheme proposed earlier we set up Hamiltonian path integral quantization for a particle in two dimensions in plane polar coordinates.This scheme uses the classical Hamiltonian, without any O(ℏ2)O(\hbar^2)O(ℏ2) terms, in the polar varivables. We show that the propagator satisfies the correct Schrödinger equation.