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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Plane-wave quantization for polygonal billiards

José Luis Vega, T. Uzer, Joseph Ford

Published Aug 1, 1995• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.52.1490

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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.
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.
We study the algorithmic complexity of motions in classical polygonal billiards, which, as the number of sides increases, tend to curved billiards, both regular and chaotic. This study unveils the equivalence of this problem to the procedure of quantization: the average complexity of symbolic trajectories in polygonal billiards features the same scaling relations (with respect to the number of sides) that govern quantum systems when a semiclassical parameter is varied. Two cases, the polygonal approximations of the circle and of the stadium, are examined in detail and are presented as paradigms of quantization of integrable and chaotic systems.
We propose a method of measuring approximate quantum eigenfunctions in polygonalized billiard geometries, based on a quasiclassical evolution operator having a (smoothened) Perron-Frobenius kernel modulated by a phase arising from quantum considerations. Using a plane wave ansatz, we show that the condition under which this is an eigenfunction of the quasiclassical operator is identical to the condition for it to be an eigenfunction of the Schrödinger equation for polygonalized billiards. Finally, we demonstrate this technique by determining the quasiclassical eigenfunctions of the polygonalized stadium billiard using arbitrary trajectories and comparing this with the exact quantum stadium eigenfunctions.
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.
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.