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Routledge eBooks

One-dimensional delta wells

P. C. W. Davies, David S. Betts

Published Apr 27, 2018• 0 citations
Publisher

Abstract

This chapter considers a particle with mass m and total energy E moving under the influence of the one-dimensional finite square well. An alternative method which turns out to be much more flexible when arrays of two or more wells are considered. The starting point of this alternative method is the boundary condition that the gradient of the wave function must be continuous at both edges of the well, and this implies that the gradient change across the well must be the same whether one uses the internal or external wave functions. Consider the array of three one-dimensional delta wells. This potential configuration can have up to three allowed states, depending on the well separation r. Since the potential is symmetrical about r = 0, the wave functions must have even or odd parity. Two of them have even parity while the third has odd parity.

DOI
10.1201/9780203741481-3

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This chapter considers a particle with mass m and total energy E moving under the influence of the one-dimensional finite square well. An alternative method which turns out to be much more flexible when arrays of two or more wells are considered. The starting point of this alternative method is the boundary condition that the gradient of the wave function must be continuous at both edges of the well, and this implies that the gradient change across the well must be the same whether one uses the internal or external wave functions. Consider the array of three one-dimensional delta wells. This potential configuration can have up to three allowed states, depending on the well separation r. Since the potential is symmetrical about r = 0, the wave functions must have even or odd parity. Two of them have even parity while the third has odd parity.
Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).