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arXiv:nlin/0301031 · Journal of Physics A Mathematical and General

Can billiard eigenstates be approximated by superpositions of plane waves?

Boris Gutkin

Published Jul 29, 2003• 11 citations
arXivHTMLPDF

Abstract

The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. From the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. In contrast, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in R 2 (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.

DOI
10.1088/0305-4470/36/32/304

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The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. From the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. In contrast, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in R 2 (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.
There is theoretical evidence indicating that generic polygons do not admit eigenfunctions that are superpositions of plane waves. However, we find that a plane-wave quantization for a family of polygonal billiards gives results in very good agreement with theoretical predictions. Indeed the fraction of missing states decreases as the number of sides is increased. We provide a resolution of this contradiction: While theory says that no plane-wave eigenfunctions are possible for polygonal boundaries, it is not applicable to the curvilinear boundaries approximating these polygons. This is analogous to the classical mechanics of polygonal billiards, where according to continuum mathematics there is no chaos, yet discrete mathematics finds it. In the quantum treatment considered here, according to continuum mathematics there can be no plane-wave eigenfunctions, yet discrete mathematics finds them.
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.
In the present article we discuss whether or not ansatzes more general than those considered hitherto, for the representation in terms of plane waves of eigensolutions of the Schrödinger equation for a free particle in a plane convex polygonal domain, lead to the construction of a complete set for a class of domains larger than the well-known one, containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves. Our conclusion is in the negative. Comments are also made on some features of a representation in terms of analytic functions of two complex variables.
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.
The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground-state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited-state theorem’’ is established which states that for every N-sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spanned by any two of these N nodal-diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and NtoinftyN \\to \\inftyNtoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.