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Procedia Engineering

Schrödinger Equations for Delta Potential Barrier in Quantum Phase Space

Jun Lu

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.

DOI
10.1016/j.proeng.2011.11.2471

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The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrödinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.