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arXiv:1809.04856

Path integrals of systems on the half-line

Seiji Sakoda

Published Sep 13, 2018• 0 citations
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Abstract

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 $1/x^{2}$.

DOI
10.48550/arxiv.1809.04856

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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.
We make use of point transformations to introduce new canonical variables for systems defined on a finite interval and on the half-line so that new position variables should take all real values from −∞-\infty−∞ to ∞\infty∞. The completeness of eigenvectors of new momentum operators enables us to formulate time sliced path integrals for such systems. Short time kernels thus obtained require extension of the range of variables to the covering space in order to take all reflected paths into account. Upon this extension we determine phase factors attached to the amplitude for paths reflected at boundaries by taking singularities of the potential into account. It will be shown that the phase factor depends on parameters that characterize the potential; and further that the well-know minus sign in the amplitude for odd times reflection of a particle in a box should be understood as the special case for the corresponding value of the parameter of the potential.
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.
We study the Feynman path-integral formalism for the constrained problem of a free particle moving on the half-line. It is shown that the effect of the boundary condition at the origin can be incorporated into the path integral by a simple modification of the action. The small-time behavior of the Green's function can be obtained from the stationary-phase evaluation of our expression for the path integral, which in this case includes contributions from both the direct and reflected classical paths.
Dfinition of the Feynman path integral in the PPP-representation