A numerical damped oscillator approach to constrained Schrödinger equations

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. We include three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the two-dimensional harmonic oscillator with degenerate excited states; and finally a non-linear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either in own 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.

arXiv commentFinal published (Open Access) version, including codes for Matlab or Octave as ancillary files

References in corpus (6)

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• Numerical solution of the stationary multicomponent nonlinear Schrödinger equation with a constraint on the angular momentum
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Cited by in corpus (4)

• Second-order flows for computing the ground states of rotating Bose-Einstein condensates
• Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions
• Stationary states of Bose-Einstein condensed atoms rotating in an asymmetric ring potential
• Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes