Solution of one-dimentional Schrodinger equation by transmission matrix method

Wayne W. Lui, Masao Fukuma

A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

S. Brandt, Hans Dieter Dahmen

Hervé Le Rouzo

A variational R-matrix method for solving the one-dimensional time-independent Schrödinger equation is presented. The theory is simple and general and applies to an arbitrary potential. As an example, the method is applied to the pedagogically important problem of quantum tunneling through a barrier. Accurate transmission and reflection coefficients are obtained by the variational R-matrix method which compare favorably with the known analytical solutions.

S. Brandt, Hans Dieter Dahmen

S. Brandt, Hans Dieter Dahmen

Petarpa Boonserm, Tritos Ngampitipan, Kunlapat Sansuk

Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.