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American Journal of Physics

Numerov extension of transparent boundary conditions for the Schrödinger equation in one dimension

Curt A. Moyer

Published Feb 12, 2004• 58 citations
Publisher

Abstract

We describe an algorithm for animating time-dependent quantum wave functions in one dimension with very high accuracy. The algorithm employs the Crank–Nicholson approximation for the time dependence along with a Numerov extension of the discrete transparent boundary conditions described recently by Ehrhardt. We illustrate the power of this approach by simulating the decay of alpha particles from radioactive nuclei and the resonance scattering of electrons in a three-layer GaAs–GaAlAs sandwich.

DOI
10.1119/1.1619141

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We describe an algorithm for animating time-dependent quantum wave functions in one dimension with very high accuracy. The algorithm employs the Crank–Nicholson approximation for the time dependence along with a Numerov extension of the discrete transparent boundary conditions described recently by Ehrhardt. We illustrate the power of this approach by simulating the decay of alpha particles from radioactive nuclei and the resonance scattering of electrons in a three-layer GaAs–GaAlAs sandwich.
We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
In this work, aiming to solve numerically the Schrödinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schrödinger equation with potentials of the form V (x) + αδp(x), where δp(x) is a pseudo-delta function, a very high and thin barrier.The numerical results show good agreement with analytical results found in the literature.Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta barrier in the center.