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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Morse-potential Green's function with path integrals

I. H. Duru

Published Nov 15, 1983• 62 citations
Publisher

Abstract

An integral representation for the Green's function of the one-dimensional Morse potential is obtained by solving path integrals. To test the method employed, the correct bound-state energy spectrum and the wave functions are derived.
DOI
10.1103/physrevd.28.2689

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On the path integration for the potential barrier V0cosh⁡−2(ωx)V_{0}\cosh ^{-2}(ωx)V0​cosh−2(ωx)
The propagator associated to the potential barrier V=V0cosh⁡−2(ωx)V=V_{0}\cosh ^{-2}(ωx)V=V0​cosh−2(ωx) is obtained by solving path integrals. The method of delta functionals based on canonical and other transformations is used to reduce the path integral for this potential into a path integral for the Morse potential problem. The dimensional extension technique is seen to be essential for performing the multiple integral representation of the propagator. The correctly normalized scattering wave functions and the scattering function are derived. To test the method employed, the free particle and the δ−δ-δ−function barrier are considered as limiting cases.
In this paper, we present analytic solutions of the Feynman propagator with the Rosen–Morse potential. To this end, an approximation of the centrifugal potential for any l state is used and nonlinear space–time transformations in the radial path integral are applied. A transformation formula that relates the original path integral to the Green function of a new quantum soluble system is derived. Explicit expressions of the bound state energy spectra and the eigenfunctions are obtained and compared to those of Schrödinger formalism. The Eckart potential is also treated as a special case.