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arXiv:physics/9702022 · International Journal of Modern Physics C

Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems

Ioan Kosztin, Klaus J. Schulten

Published Apr 1, 1997• 42 citations
arXivHTMLPDF

Abstract

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

DOI
10.1142/s0129183197000278

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The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.
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 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.
Introduction: Nowadays extraordinary properties of nano world and economical problems are the reasons for directing techniques toward micro implements, electronic circuits dimensions have been reduced to 10 to 100 nano meters. Moreover microstructures in cells with nano meter dimensions have been investigated by biologists, there are structures in electronic and biology which are very similar to the billiards and the electromagnetic waveguides but their dimensions are in nano meter range, therefore they were called quantum waveguides and billiards. Aim: In this research billiards and quantum waveguides at any arbitrary cross sectional area have been investigated. Method: A considerable effort was devoted in solving Helmholtz’s equation with two dimensional cross sectional areas. Linear combinations of plane waves or Heller methods for several types of connective and no connective billiards with regular and irregular shapes have been used. Then eigenvalues and eigenvectors were calculated. At the final part by applying a simple method billiard type (chaotic or quiet) in a quantum scale has been realized. Results and Discussions: In this paper two types of circular and stadium billiards have been studied and by using Heller’s method and applying the numerical solution, energy levels together with the function forms have been attained. Conclusions: This paper shows that Heller’s method despite its quickness and simplicity is a very accurate method for finding eigenvalues and eigenvectors. Although in high eigenvalues the calculation is weighted, nevertheless this method is an acceptable computational method for the investigation of quantum nanobilliards such as quantum points.
We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate the existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits Gaussian Unitary Ensemble (GUE) like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunnelling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.
We examine the quantum mechanical eigensolutions of the two-dimensional infinite-well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of ℏ/2. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, N ( E ), comparing this system to the standard circular well. These two billiard geometries have the same area ( A = π R 2 ), but different perimeters ( P = 2π R versus (2π + 2) R ), and we compare both cases to fits of N ( E ) which make use of purely geometric arguments involving only A and P . We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite-well plus δ-function potential.