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arXiv:hep-th/9303128 · Physical Review Letters

Path integration via summation of perturbation expansions and applications to totally reflecting boundaries, and potential steps

Ch. Grosche

Published Jul 5, 1993• 83 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.71.1

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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.
The path integral treatment of the step potential based on exact summation of the Feynman perturbation series is presented in its original form. With a two-sided Laplace transform on the end point, we have established a recurrence formula between three successive terms of the perturbation series which leads us to sum up exactly all the terms and deduce the Green function of the problem. The Wiener–Hopf method and Hilbert problems are successfully used for solving our problem with the boundary conditions.
Abstract A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half‐spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of δ‐function perturbations is outlined, which includes the discussion of multiple δ‐function perturbations, δ‐function perturbations along perpendicular lines and planes, and moving δ‐function perturbations. The limiting process, where the strength of the δ‐function perturbations gets infinitely repulsive, has the effect of producing impenetrable walls at the locations of the δ‐function perturbations, i.e. a consistent description for boundary problems with Dirichlet boundary‐condition emerges. Several examples illustrate the formalism.
delta '-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is consideration of the path integral representation for the one-dimensional Dirac particle together with a relativistic point interaction. The nonrelativistic limit yields either the usual delta -function or a delta '-function perturbation; making their strengths infinitely repulsive one respectively obtains either Dirichlet or Neumann boundary conditions in the path integral.
#delta#'-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual #delta#-function or a #delta#'-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. (orig.)
A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ∫d3rβK(rβ,0), where K(rβ,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a ``modified'' perturbation expansion (W is expanded as a power series in β with β = 1/kT). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.