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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum mechanics on the half-line using path integrals

T. E. Clark, Ralph Menikoff, D. H. Sharp

Published Dec 15, 1980• 79 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.22.3012

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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.
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.
AbstractThis work investigates a variational perspective on the Feynman path integral, aiming to explore connections between stationary action, probability amplitude, integrating over path evolutions, and the semiclassical limit. The variational analysis employed should be viewed as conceptually explorative and heuristic rather than strictly formal. It explores how variations of path evolutions in the path integral at the level of a single infinitesimal slice can yield meaningful insight into the behavior of quantum trajectories. A local variational analysis of the free particle and a particle near a constant potential provides insight into how the stationary-action principle underlies both classical and quantum behavior, through a local relationship between the classical slice-action and a functional over the phase-weighting of quantum path evolutions. In particular, this functional is proportional to the slice-action itself up to additive and multiplicative constants. This proportionality implies that the stationarity of this functional is equivalent to the stationarity of the slice-action, which in turn generates the semiclassical stationary path evolution. The resulting formulation produces results consistent with known short-time propagators for the free particle and a particle near a constant potential, including tunneling behavior.
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.
A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.