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College Physics

A reverse algorithm for one- dimensional quantum scattering

Dong Jia

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

A reverse algorithm is presented for computation of one- dimensional quantum scattering. It is assumed that there is only the transmitted wave in transmission side. The Schrdinger equation is solved from the transmission side to the incident side according to the initial value problem,the real part and the imaginary part of the wave function,probability density function and then in the incident side,the maximum and minimum value of the function are obtained,the transmittance and reflectance arealso calculated simply. Variation of the quantum phase in scattering process is studied by using this method,an interesting phenomenon of π- jump's in the reflection wave phase at transparent points is found.

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การหาขอบเขตลางของความนาจะเปนของการสงผานในปญหาการกระเจงทางควอนตมใน 1 มต

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A reverse algorithm is presented for computation of one- dimensional quantum scattering. It is assumed that there is only the transmitted wave in transmission side. The Schrdinger equation is solved from the transmission side to the incident side according to the initial value problem,the real part and the imaginary part of the wave function,probability density function and then in the incident side,the maximum and minimum value of the function are obtained,the transmittance and reflectance arealso calculated simply. Variation of the quantum phase in scattering process is studied by using this method,an interesting phenomenon of π- jump's in the reflection wave phase at transparent points is found.
One-dimensional scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. First, the pole structure of the quantum momentum function for scattering wave functions is analyzed. The significant differences of the pole structure of this function between scattering wave functions and bound state wave functions are pointed out. An accurate computational method for the quantum Hamilton-Jacobi equation for general one-dimensional scattering problems is presented to obtain the scattering wave function and the reflection and transmission coefficients. The computational approach is demonstrated by analysis of scattering from a one-dimensional potential barrier. We not only present an alternative approach to the numerical solution of the wave function and the reflection and transmission coefficients but also provide a computational aspect within the quantum Hamilton-Jacobi formalism. The method proposed here should be useful for general one-dimensional scattering problems.
Quantum mechanics is the theory that describes dynamics of small objects such as atom and molecule. In this paper, Schrodinger’s wave mechanics, a part of quantum mechanics, is studied. The central equation of this wave mechanics is the Schrodinger’s equation. Solving this equation, quantum system dynamics can be described. In this work, the quantum scattering problem in one dimension is studied. Wave functions are obtained by exactly solving the Schrodinger’s equation in case of the delta function potential and the rectangular potential. The transmission and reflection probabilities are calculated from the obtained wave functions. Lower bounds on the transmission probabilities are presented. Finally, the lower bounds on the transmission probabilities are applied to the delta function potential and the rectangular potential problems. The results show that the exact transmission probabilities satisfy the lower bounds on the transmission probabilities.
A new numerical algorithm for tunneling is presented. This algorithm can be used to analyse an onedimensional\npotential barrier with an arbitrary profile. It is able to calculate not only the transmission coefficient\naccurately, but also the phases of the transmitted and reflected waves. Using the Schrodinger equation and plane waves,\nwe introduce a tunneling matrix to relate the transmitted and incident wave functions. Thus the transmission coefficient\nand phase can be calculated from the tunneling matrix.To test the accuracy of this algorithm,we calculate the transmission\ncoefficients and phases for a double -barrier (resonant tunneling) and a triangular barrier (field emission), and\nmake comparisous with corresponding analytical values. It shows that this algorithm is fast, accurate and efficient.
The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.