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arXiv:2310.13166

The semiclassical states excitations in the multi-rectangular billiards

Stefan Giller

Published Oct 19, 2023• 0 citations
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Abstract

The problem of the quantizations of the $L$-shaped billiards and the like ones, i.e. each angle of which is equal to $π/2$ or $3π/2$, is considered using as a tool the Fourier series expansion method. The respective wave functions and the quantization conditions are written and discussed looking for and discussing about the superscars effects in such multi-rectangular billiards (MRB). It is found that a special set of POC modes effect the superscars phenomena in MRB in which the billiards are excited as a whole to the modes closest to the semiclassical ones existing in their approximated copies being MRB in which their parallel sides remain in rational relations between themselves.

DOI
10.48550/arxiv.2310.13166

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Superscars are quantum eigenstates that are localized in an extended region of a family of parallel periodic orbits. They are prevalent in pseudointegrable systems. Here we consider a pseudo-integrable right-triangular quantum billiard with an acute angle π/5 as an example. In the configuration space, we identify and explain a fringe structure of superscar eigenstates originating from a correlation between the two quantum numbers imposed by the sign change rule of the wave function during unfolding. In the wave-vector space, we unveil unique characteristics of the peaks for superscar states, and in return, exploit these features to determine the family of periodic orbits and the corresponding quantum numbers without ambiguity. We also find that the eigenstates can be superpositions not only of superscar modes corresponding to different families of periodic orbits, but also of superscar modes corresponding to the same family but with different sets of quantum numbers, which poses a challenge for conventional approaches in identifying the superscar modes but can be addressed properly by our analysis. These results are also applicable to superscar states in other pseudointegrable systems.
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.
The eigenvalues of the Hyperspherical billiard are calculated in the semiclassical approximation. The eigenvalues where this approximation fails are identified and found to be related to caustics that approach the wall of the billiard. The fraction of energy levels for which the semiclassical error is larger than some given value is calculated analytically (and tested numerically) and found to be independent of energy. The implications for other systems, in particular integrable ones, are discussed.
The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and