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Journal of Physics A Mathematical and General

Exact propagator for reflectionless potentials

Richard E. Crandall

Published Sep 11, 1983• 22 citations
Publisher

Abstract

The exact space-time propagation for a general reflectionless potential is obtained in closed form involving error functions. It is indicated how bound states and transmission coefficients may be recovered from asymptotic behaviour of the propagator. A sum rule is derived that shows how the leading terms of the short-time form of the propagator can be used rigorously in a Feynman path integral formalism.

DOI
10.1088/0305-4470/16/13/023

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The exact space-time propagation for a general reflectionless potential is obtained in closed form involving error functions. It is indicated how bound states and transmission coefficients may be recovered from asymptotic behaviour of the propagator. A sum rule is derived that shows how the leading terms of the short-time form of the propagator can be used rigorously in a Feynman path integral formalism.
An analytic form for the nonrelativistic Coulomb propagator is derived, thus resolving a long-standing problem in Feynman's path-integral formulation of quantum mechanics. Hostler's formula for the Coulomb Green's function is expanded according to the theorem of Mittag-Leffler, then Fourier transformed term by term to give the Coulomb propagator. The result is a discrete summation over the principal quantum number n, involving Whittaker, Laguerre, Hermite, and error functions. As is the case for other nonquadratic potentials, the Coulomb propagator does not have the canonical structure K=F exp(iS/\ensuremath{\Elzxh}). Part of the expansion resembles a form derived by Crandall [J. Phys. A 16, 3005 (1983)] for the case of reflectionless potentials.
We will derive a rigorous real-time propagator for the non-relativistic quantum mechanical L 2 transition probability amplitude and for the non-relativistic wavefunction. The propagator will be given explicitly in terms of the time evolution operator. The derivation will be for all self-adjoint non-vector potential Hamiltonians. For systems with potentials that carry at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real-time, time-sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in a non-standard analysis formulation. Finally, we will compute the propagator for the harmonic oscillator using the non-standard analysis Feynman path-integral formulation; we will compute the propagator without using any knowledge of the classical properties of the harmonic oscillator.
Let GGG be the time-dependent quantum mechanical propagator for a particle that is free except for a half-plane barrier with a straight edge. A closed-form exact expression for GGG is presented in terms of the Fresnel integral for the case where the initial and final positions lie in a plane perpendicular to the edge. GGG is not expressible as a sum over classical paths.
The authors derive a closed-form expression for the time-dependent propagator for a quantum mechanical particle which is subject to an external force which is the sum of (i) a reflecting half-plane barrier with a straight edge, and (ii) a harmonic force pointing towards a point of the edge. This new addition to the short list of exactly known quantum mechanical Green functions is a simple combination of exponential functions and Fresnel integrals, the arguments of which are combinations of trigonometric functions.