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.
Highlights
In this article we describe the idea of solving stationary Schrödinger equations (SEs) as energy minimization problems with constraints, by using a second-order damped dynamical system
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
We have introduced the dynamical functional particle method (DFPM) with normalization and several orthogonalization constraints for the linear SE
Summary
In this article we describe the idea of solving stationary Schrödinger equations (SEs) as energy minimization problems with constraints, by using a second-order damped dynamical system. -called imaginary time dependent SE (see below), the approach is, after discretization in space (finite differences, finite elements, or other methods), to solve a first-order damped time dependent equation numerically, see [1, 2] Sometimes these methods are called steepest descent methods (not to be confused with steepest descent methods in optimization [3]). Our method presented here can be used in a very general and efficient way for minimization problems (linear, nonlinear, with any type of global constraints) and has been shown to be highly competitive [7,8,9] It can be used for linear eigenvalue problems [10, 11]. We hope the readers will expand the theory and applications in different directions from the examples presented here
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