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arXiv:0910.0574 · American Journal of Physics

Numerical approach to the Schrödinger equation in momentum space

William A. Karr, Christopher R. Jamell, Yogesh N. Joglekar

Published Mar 11, 2010• 8 citations
arXivHTMLPDF

Abstract

The treatment of the time-independent Schrödinger equation in real space is an indispensable part of introductory quantum mechanics. In contrast, the Schrödinger equation in momentum space is an integral equation that is not readily amenable to an analytical solution and is rarely taught. We present a numerical approach to the Schrödinger equation in momentum space. After a suitable discretization process, we obtain the Hamiltonian matrix and diagonalize it numerically. By considering a few examples, we show that this approach is ideal for exploring bound states in a localized potential and complements the traditional (analytical or numerical) treatment of the Schrödinger equation in real space.

DOI
10.1119/1.3272021

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The treatment of the time-independent Schrödinger equation in real space is an indispensable part of introductory quantum mechanics. In contrast, the Schrödinger equation in momentum space is an integral equation that is not readily amenable to an analytical solution and is rarely taught. We present a numerical approach to the Schrödinger equation in momentum space. After a suitable discretization process, we obtain the Hamiltonian matrix and diagonalize it numerically. By considering a few examples, we show that this approach is ideal for exploring bound states in a localized potential and complements the traditional (analytical or numerical) treatment of the Schrödinger equation in real space.
We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.
Abstract This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the 2D harmonic oscillator with degenerate excited states; and a nonlinear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either with coding or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can, for example, use our derived results and the code (supplemental material ( https://stacks.iop.org/EJP/41/065406/mmedia )) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.
This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the 2D harmonic oscillator with degenerate excited states; and a nonlinear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either with coding or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can, for example, use our derived results and the code (supplemental material (https://stacks.iop.org/EJP/41/065406/mmedia)) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.