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Journal of Mathematical Physics

The polygon quantum-billiard problem

Richard L. Liboff

Published Feb 1, 1994• 32 citations
Publisher

Abstract

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.

DOI
10.1063/1.530874

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•Jul 9, 2003•20 citations
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.
Stemming from known properties of one-dimensional (1-D) and 2-D quantum billiards, it is conjectured that the nodal surface of the first-excited state of the convex 3-D quantum billiard intersects the billiard surface in a single simple closed curve. Examples of the validity of this conjecture are given for a number of elementary 3-D billiard configurations. From these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation. Two characteristic displacement parameters are defined which are labeled an axis length, L, and diameter, a. It is conjectured that a parameter κ≊1, exists, whose exact value depends on the properties of the billiard, such that for L≳κa (‘‘prolatelike’’) the nodal surface of the first-excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half. For L<κa (‘‘oblatelike’’) the nodal surface of the first-excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half. Arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards, termed ‘‘spherical-like.’’ It is hypothesized that the nodal surface of the first-excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron.
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 consider the radially vibrating spherical quantum billiard as a representative example of vibrating quantum billiards. We derive necessary conditions for quantum chaos in ddd-term superposition states. These conditions are symmetry relations corresponding to the relative quantum numbers of eigenstates considered pairwise. In this discussion, we give special attention to eigenstates with null angular momentum (for which the aforementioned condition is automatically satisfied). Moreover, when these necessary conditions are met, we observe numerically that there always exist parameter values that correspond to chaotic configurations. We focus our numerical studies on the ground and first excited states of the radially vibrating spherical billiard with null angular momentum eigenstates. We observe chaotic behavior in this configuration, which dispels the common belief that one is required to pass to the semiclassical (ℏ⟶0\hbar \longrightarrow 0ℏ⟶0) or high quantum number limits in order to meaningfully discuss quantum chaos. The results in the present paper are also of practical import, as the radially vibrating spherical quantum billiard may be used as a model for the quantum dot nanostructure, the Fermi accelerating sphere, and intra-nuclear particle behavior.
An electron in quantum confinement takes on a discrete energy spectrum which is defined based on the solution to the Schrodinger Equation for a given potential. Well defined closed-form energy spectra are known for the particle in a box, circular potential, quarter circle potential, and an equilateral triangle. A closed-form solution for more complex shapes may not be known, but numerical methods can be used to find an approximate solution. In this research, an application of the Finite Element Method (FEM) in Wolfram Mathematica is presented and applied to Quantum Billiards with a variety of geometries. To assess the accuracy of the method, the computed energy states are analyzed in the limit of a polygon with an increasing number of sides, the numerical results are validated against analytical solutions for geometries with known exact forms, and a standard convergence test is conducted. The FEM results closely match analytical solutions for known potentials, demonstrating its high accuracy. For high energy index n, quantum scarring may emerge for certain geometries. The nature of quantum scarring and its presence in the computed models is also investigated qualitatively.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger 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 N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions 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 wave packets 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, quasiclassical 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.
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.