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arXiv:1809.01732

Decomposition of the Feynman kernel for a particle in a box

Seiji Sakoda

Published Sep 3, 2018• 0 citations
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Abstract

We study the decomposition of the Feynman kernel for a particle in a box with $1/\sin^{2}θ$ potential to find that the wellknown phase factor $-1$, which is correct for the case of the free particle, for reflection at boundaries should be generalized depending on the parameter of the potential.

DOI
10.48550/arxiv.1809.01732

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We study the decomposition of the Feynman kernel for a particle in a box with 1/sin⁡2θ1/\sin^{2}θ1/sin2θ potential to find that the wellknown phase factor −1-1−1, which is correct for the case of the free particle, for reflection at boundaries should be generalized depending on the parameter of the potential.
It is wellknown that the Feynman kernel for the free particle on the half-line can be expressed as a sum over classical paths if we take the contribution from the reflected path into account. The minus sign for the reflected path needs to be generalized depending on the parameter of the potential if we add a potential proportional to 1/x21/x^{2}1/x2.
The propagator of a particle in a one-dimensional box is quite different from the propagator for infinite space. It has a complicated but regular structure of everywhere dense singularities. We investigate the case of periodic boundary conditions both numerically and theoretically, leaving the slightly more complicated case of reflecting boundary conditions to the suggested problems. The singularity structure is shown to match up with the fractional revivals observed by many authors in wave packet studies.