On the solution of the nonstationary Schr\"odinger equation

E. Yu. Mishcharina ; E. E. Libin ; M. A. Bubenchikov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2016), pp. 28-34

Résumé

The Schrödinger equation describes quantum mechanics processes occurring when particles pass through a potential barrier. In this problem, it is necessary to find the probability density of particles and to track its evolution in time. In this paper, it is shown that time-dependent Schrödinger's equation has a direct analogy to the heat conductivity equation, differing from it in the imaginary time. As a numerical method of the decision, it is offered to apply the method of matrix exponential function in which a finite difference analogue of the one-dimensional Laplacian is considered as a matrix operating on a vector. This way of the solution allows one to consider potential barriers of any form in the Schrödinger equation. Time is included now into the decision as a parameter, and it allows one to get rid of the necessity of time quantization and to do it only on a spatial variable. In this aspect, this way favorably differs from traditional ways of solving evolutionary equations which use quantization both on time and on a spatial variable. Results of numerical experiments show that the greatest amplitudes of probability are localized in the field of minima of potential barriers.

Détail

Keywords: probability amplitude, Schrödinger equation, matrix exhibitor, heat conductivity equation, potential barriers.