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Physical review. B, Condensed matter

Semiclassical analysis of response functions for integrable systems

B. Mehlig

Published Apr 15, 1997• 5 citations
Publisher

Abstract

We derive a semiclassical expansion of quantum spectral functions for integrable systems. This complements a previous analysis of chaotic billiards [B. Mehlig et al., Phys. Rev. Lett. 75, 57 (1995)]. We discuss similarities and differences between the integrable and the chaotic case. As a numerical example, we consider a two-dimensional circular billiard.

DOI
10.1103/physrevb.55.r10193

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We derive a semiclassical expansion of quantum spectral functions for integrable systems. This complements a previous analysis of chaotic billiards [B. Mehlig et al., Phys. Rev. Lett. 75, 57 (1995)]. We discuss similarities and differences between the integrable and the chaotic case. As a numerical example, we consider a two-dimensional circular billiard.
The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
Using a recently defined quantum spectral function and the method of closed-orbit theory we studied the spectra of the half-circular and the quarter circular billiard systems for arbitrary two points.The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in well.Our calculation demonstrates that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits.The two examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.
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.
The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.
This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.