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arXiv:1303.3848 · Physical Review E

Semiclassical wave functions for open quantum billiards

Fabian Lackner, Iva Březinová, Joachim Burgdörfer, Florian Libisch

Published Aug 19, 2013• 13 citations
arXivHTMLPDF

Abstract

We present a semiclassical approximation to the scattering wave function Ψ(r,k) for an open quantum billiard, which is based on the reconstruction of the Feynman path integral. We demonstrate its remarkable numerical accuracy for the open rectangular billiard and show that the convergence of the semiclassical wave function to the full quantum state is controlled by the mean path length or equivalently the dwell time for a given scattering state. In the numerical implementation a cutoff length in the maximum path length or, equivalently, a maximum dwell time τ(max) included implies a finite energy resolution ΔE~τ(max)(-1). Possible applications include leaky billiards and systems with decoherence present.

DOI
10.1103/physreve.88.022916

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We present a semiclassical approximation to the scattering wave function Ψ(r,k) for an open quantum billiard, which is based on the reconstruction of the Feynman path integral. We demonstrate its remarkable numerical accuracy for the open rectangular billiard and show that the convergence of the semiclassical wave function to the full quantum state is controlled by the mean path length or equivalently the dwell time for a given scattering state. In the numerical implementation a cutoff length in the maximum path length or, equivalently, a maximum dwell time τ(max) included implies a finite energy resolution ΔE~τ(max)(-1). Possible applications include leaky billiards and systems with decoherence present.
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.
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.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
Quantum billiards has attracted much interest recent years.There are many works on the two-dimensional billiards(2D).Compare with the 2D billiards,the three-dimensional(3D) billiards approaches with the actual system better.Taking the cubic billiards as an example,using the closed orbit theory(COT),we find the classical open orbits and study the correspondence between the quantum spectra and the length of classical open orbits in the 3D billiard system.The peaks' locations of the quantum spectra match with the length of the classical open orbits very well.This will help us to analyze the transport property through an open billiards.By this method,the physical image is clear,the computation is easier and it can help us to understand the character of some chaotic systems better.This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.