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Journal of Physics A Mathematical and General

A new exact quantum mechanical propagator

F.W. Wiegel, P W van Andel

Published Feb 21, 1987• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/3/025

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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.
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 new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.
We study the time-dependent propagator for an electron which moves in a plane and which is subject to quadrupolar electric fields with certain reflection symmetries, and to a uniform magnetic field perpendicular to the plane. It is shown that the electric potential, which has a saddle point at the origin, can be produced by voltages applied to four symmetrically configured cylindrical electrodes whose axes are perpendicular to the plane. We use path-integral methods to derive a closed-form expression for the propagator, and comment in some detail on the structure of its normalization factor.