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Journal of Physics A Mathematical and Theoretical

Explicit expression for the reflection and transmission probabilities through an arbitrary potential barrier

Pengyi Su, Zhuangqi Cao, Kai-Sheng Chen, Cheng Yin, Qishun Shen

Published Oct 16, 2008• 9 citations
Publisher

Abstract

Without solving the Schrödinger equation, exact and general expressions of the reflection and transmission probabilities for a quantum particle through an arbitrary potential barrier are presented by using the analytical transfer matrix method. It is seen that in addition to the parameters of the ambience and both the start and the final point of the barrier, the unique dependence on the formulae is the total phase shift accumulated by the mainwaves and the subwaves.

DOI
10.1088/1751-8113/41/46/465301

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Without solving the Schrödinger equation, exact and general expressions of the reflection and transmission probabilities for a quantum particle through an arbitrary potential barrier are presented by using the analytical transfer matrix method. It is seen that in addition to the parameters of the ambience and both the start and the final point of the barrier, the unique dependence on the formulae is the total phase shift accumulated by the mainwaves and the subwaves.
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
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.
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.
Based on the transmission formula derived from the analytical transfer matrix method,two cases of the transmission probabilities of the position-dependent-mass particle through the free potential field are calculated.Combined with the point canonical transformation,we conclude that the transmission probabilities are closely connected to the structure of their effective potential.If the effective potential is a well,the transmission resonance peaks lies exactly at the eigen-energies.If the effective potential is a barrier,the transmission probabilities are very low at the below-barrier energies and oscillating around 1 at the beyond-barrier energies.
We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.