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Physical Review A

Amplitude ordering of the trace formula for the two-particle disk billiard

Jamal Sakhr

Published Sep 10, 2001• 0 citations
Publisher

Abstract

For the periodic orbit quantization of a system of particles confined in a two-dimensional disk billiard, the proliferation of multiparticle periodic orbits with length precludes standard length truncation of the trace formula and alternative truncation schemes are necessary. An amplitude ordering of the single-particle periodic orbits reveals a quantum-classical correspondence for the angular momentum of a particle in the disk which is useful for the quantization of all the low energy states of the multiparticle problem. We explicitly show that a simple amplitude truncation of the trace formula better approximates the semiclassical density of states and in particular resolves closely spaced levels that are otherwise unresolved using the standard approach.

DOI
10.1103/physreva.64.044102

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For the periodic orbit quantization of a system of particles confined in a two-dimensional disk billiard, the proliferation of multiparticle periodic orbits with length precludes standard length truncation of the trace formula and alternative truncation schemes are necessary. An amplitude ordering of the single-particle periodic orbits reveals a quantum-classical correspondence for the angular momentum of a particle in the disk which is useful for the quantization of all the low energy states of the multiparticle problem. We explicitly show that a simple amplitude truncation of the trace formula better approximates the semiclassical density of states and in particular resolves closely spaced levels that are otherwise unresolved using the standard approach.
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The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.
Using Feynman path integrals and the stationary-phase method, we develop a semiclassical theory for quantum trace formulas in classically hyperbolic systems. In this way, we obtain corrections to the Gutzwiller-Selberg trace formula as an asymptotic series in powers of the Planck constant. The first coefficient of this series is given explicitly. We illustrate the theory with the calculation of complex-wave-number resonances for the two-disk scatterer and show that effects beyond the Gutzwiller leading approximation are at the origin of a lengthening of the resonance lifetimes at low energy.
Abstract One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for <?CDATA N=2,3N = 2,3N=2,3?> N = 2 , 3 and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.
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
Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.