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arXiv:quant-ph/0302009 · NCimB

On the path integration for the potential barrier $V_{0}\cosh ^{-2}(ωx)$

Larbi Guechi, Théophile F. Hammann

Published Feb 2, 2003• 0 citations
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Abstract

The propagator associated to the potential barrier $V=V_{0}\cosh ^{-2}(ωx)$ 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 $δ-$function barrier are considered as limiting cases.

DOI
10.48550/arxiv.quant-ph/0302009

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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.
It is proposed that each of several analytical techniques used to evaluate the path integral when a spinless particle encounters an infinite potential barrier relies on more constraints than is necessary. We review the image, forbidden-paths, and eigenfunction-expansion methods, and the infinitesimal-propagator method is developed herein. It is then shown that the latter two methods contain assumptions leading to extra terms in the path-integral integrand which serve only to aid in the analytical evaluation of the path integral but which do not affect the outcome of that evaluation. The analytical arguments are further confirmed by a recently developed numerical method that, for a limited number of cases, is a direct evaluation of the path integral.
Path integral over single-sheeted hyperboloid is evaluated. To relate the obtained propagator to quantum system we choose the part of it which is invariant with respect to reflection. This part corresponds to the propagator of the free particle motion on the invariant subspace of M- Imaginary Lobachevsky space. The wave functions and spectra for the quantum systems with potentials -\frac{\nu^{2}+\frac{1}{4}}{\sin h^{2}x} and -\frac{\nu^{2}+\frac{1}{4}}{\sin^{2}x} is given. Note that these potentials are not the special case of Rosen-Morse or the modified P\"{o}schl-Teller ones