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arXiv:1001.2580 · American Journal of Physics

Solution of the quantum initial value problem with transparent boundary conditions

Alejandro Puga, Bruce N. Miller

Published Dec 20, 2012• 2 citations
arXivHTMLPDF

Abstract

Many systems of interest at the forefront of technological development, for example, in quantum computation, consist of weakly interacting elements that obey quantum mechanics. A challenge for modern theoretical physics is to develop a systematic methodology for approaching these “open” quantum systems. In particular, applied mathematicians have formulated the method of transparent boundary conditions (TBCs) to model interacting quantum systems of infinite extent. Here, we consider a particular application of TBCs to the escape of a particle from a one-dimensional infinite well when one of the bounding barriers is extinguished. We analytically obtain the exact time-dependent wave function by a Green's function method and then show that a numerical solution based on the TBC method is in excellent agreement. This work was motivated by an experiment carried out at the University of Texas on escape from a gravitational wedge billiard. Physicists have used billiards to understand and explore both classical and quantum chaos. Although the original experiment was carried out in the classical regime, current work is probing lower temperatures, where a quantum mechanical formulation is required.

DOI
10.1119/1.4762859

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Many systems of interest at the forefront of technological development, for example, in quantum computation, consist of weakly interacting elements that obey quantum mechanics. A challenge for modern theoretical physics is to develop a systematic methodology for approaching these “open” quantum systems. In particular, applied mathematicians have formulated the method of transparent boundary conditions (TBCs) to model interacting quantum systems of infinite extent. Here, we consider a particular application of TBCs to the escape of a particle from a one-dimensional infinite well when one of the bounding barriers is extinguished. We analytically obtain the exact time-dependent wave function by a Green's function method and then show that a numerical solution based on the TBC method is in excellent agreement. This work was motivated by an experiment carried out at the University of Texas on escape from a gravitational wedge billiard. Physicists have used billiards to understand and explore both classical and quantum chaos. Although the original experiment was carried out in the classical regime, current work is probing lower temperatures, where a quantum mechanical formulation is required.
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
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
We introduce an additional method to solve Schr\"odinger's equation for a free particle in an infinite well of arbitrary shape (the Helmholtz equation with Dirichlet boundary conditions), a problem of interest in the area of quantum chaos. We expand the wave function in a basis of products of sine functions, then use the constraint operator to contain the wave function to a region within the domain of the basis functions. In this manner, a quantum billiard problem of arbitrary shape can be solved. Several methods exist to solve problems of this sort, but as recent work reviewing these methods has shown, all have shortcomings. Our work represents a different direction in the solution of these problems. Our method is different in that it provides a means of computing an eigenbasis. It is also interesting from a physical standpoint in that it can represent the Hamiltonian of a classically chaotic system in the basis of a classically regular system. \textcopyright{} 1996 The American Physical Society.
We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore, they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limaçon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.
In this paper an exact transparent boundary condition for the multidimensional Schrödinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for two-dimensional (2D) and 3D equations reported before. An exact fully discrete (i.e., derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two noninteracting particles is considered. The proposed boundary condition is simple, robust, and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schrödinger equation (including multiparticle problems) is required.
A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.