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Physical Review A

Exact solution to the time-dependent Schrödinger equation in two dimensions

R. S. Kaushal

Published Sep 1, 1992• 13 citations
Publisher

Abstract

With a view to obtaining an exact analytic solution to the time-dependent Schr\"odinger wave equation for noncentral harmonic and anharmonic potentials in two dimensions, we make use of an ansatz for the eigenfunction in conjunction with the methods of Burgan et al. [Phys. Lett. 74A, 11 (1979)] and Ray [Phys. Rev. A 26, 729 (1982)]. While the method is found to work well for a restricted class of harmonic potentials, its limitations in the case of anharmonic potentials are pointed out.

DOI
10.1103/physreva.46.2941

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With a view to obtaining an exact analytic solution to the time-dependent Schr"odinger wave equation for noncentral harmonic and anharmonic potentials in two dimensions, we make use of an ansatz for the eigenfunction in conjunction with the methods of Burgan et al. [Phys. Lett. 74A, 11 (1979)] and Ray [Phys. Rev. A 26, 729 (1982)]. While the method is found to work well for a restricted class of harmonic potentials, its limitations in the case of anharmonic potentials are pointed out.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schrödinger equation, which is given by the Airy function. The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation. Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential. The method is shown to work in both one and two dimensions, and can be easily extended if required. The applicability of the method is discussed in relation to the nonlinear case.