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arXiv:0901.3454 · Journal of Physics Conference Series

The propagator for the step potential using the path decomposition expansion

James M. Yearsley

Published Jun 1, 2009• 2 citations
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Abstract

We present a direct path integral derivation of the propagator in the presence of a step potential. The derivation makes use of the Path Decomposition Expansion (PDX), and also of the definition of the propagator as a limit of lattice paths.
DOI
10.1088/1742-6596/174/1/012072

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The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.
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