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arXiv:nlin/0107055 · Conference Publications

The Radially Vibrating Spherical Quantum Billiard

Mason Alexander Porter, Richard L. Liboff

Published Jul 24, 2001• 2 citations
arXivHTMLPDF

Abstract

We consider the radially vibrating spherical quantum billiard as a representative example of vibrating quantum billiards. We derive necessary conditions for quantum chaos in $d$-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 ($\hbar \longrightarrow 0$) 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.

DOI
10.48550/arxiv.nlin/0107055

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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.
Eigenstates of a particle confined to a cone of finite length capped by a spherical surface element are derived. A countable infinite set of solutions is obtained corresponding to integer azimuthal and orbital quantum numbers ( m , l ) \left ( m, l \right ) . These solutions apply to a discrete subset of the domain of half vertex angles, 0 ≤ θ 0 ≤ π / 2 0 \le {\theta _0} \le \pi /2 . For arbitrary real orbital quantum numbers, l → ν l \to \nu , solutions are given in terms of the hypergeometric function, with ν = ν ( θ 0 ) \nu = \nu \left ( {\theta _0} \right ) , and are valid in the θ 0 {\theta _0} domain, 0 ≤ θ 0 > π / 2 0 \le {\theta _0} > \pi /2 . Eigenstates are either nondegenerate or two-fold degenerate. Numerical examples of both classes of solutions are included. For the case μ = cos ⁡ π / 4 \mu = \cos \pi /4 , the ground-state wavefunction and eigenenergy are \[ φ G
Abstract According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.
Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments. Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].
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.
Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.