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Spectrum guided integration for nonrelativistic quantum-mechanical problems

Thomas Gläßle

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The numerical integration of the Schrödinger equation is an enduring challenge for modern computer algorithms. Today, many numerical schemes with different numerical properties are available for its solution. A complementary approach is to alter the physical description of the problem to improve the numeric performance of the used integration algorithm. The spectrum-guided-integration utilizes a gauge transformation to render the wave function easier to master for numerical algorithms. This method has proven to be of particular value if the quantum system behaves much like its classical analogon. In this thesis the technique is generalized for usage on systems that may feature strong quantum effects. An estimate of the error improvement is discussed. To illustrate and check the considerations, the methods are applied in three model systems. These are the harmonic oscillator, a laser interaction with a bound state in a softcore potential and a scattering event with a central softcore potential.

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The numerical integration of the Schrödinger equation is an enduring challenge for modern computer algorithms. Today, many numerical schemes with different numerical properties are available for its solution. A complementary approach is to alter the physical description of the problem to improve the numeric performance of the used integration algorithm. The spectrum-guided-integration utilizes a gauge transformation to render the wave function easier to master for numerical algorithms. This method has proven to be of particular value if the quantum system behaves much like its classical analogon. In this thesis the technique is generalized for usage on systems that may feature strong quantum effects. An estimate of the error improvement is discussed. To illustrate and check the considerations, the methods are applied in three model systems. These are the harmonic oscillator, a laser interaction with a bound state in a softcore potential and a scattering event with a central softcore potential.
The integration of time-dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The wave-function's energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations. We demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude. Our approach includes the so-called Kramers-Henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency.
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.
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville’s theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.
Chapter Preview: We begin at the beginning with Plank's black-body radiation spectrum applied to the cosmic background. We move on to bound states; first finding them semi-analytically with a trial-and-error search, and then finding them numerically as an eigenvalue problem. We include relativistic effects in hydrogen bound states by solving the Klein-Gordon equation. We then examine the numerical solution of the time- independent Schrödinger equation, and various types of bound state wave functions. We move on to wave packet solutions of the time- dependent Schrödinger equation. Then we study scattering from: a 3-D square well, the Coulomb potential, three stationary disks, and on a billiard table (see also classical billiards in § 3.8 ). We next look at quantum mechanics in momentum space: integral equations, bound states, the delta-shell potential, hyperfine structure of hydrogen, SU3 symmetry of quarks. Next to last we examine several examples of coherent states. We end the chapter with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm. If not enough, Chapter 8 discussion of quantum entanglement fits in well here too.
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.