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physica status solidi (b)

Quantum-Classical Correspondence for Two Interacting Particles in a One-Dimensional Box

Lilia Meza-Montes, Felix M. Izrailev, Sergio E. Ulloa

Published Jul 1, 2000• 7 citations
Publisher

Abstract

We study the model of two interacting particles moving in a 1D box, paying main attention to the quantum-classical correspondence for the average shape of quantum eigenstates and for the local density of states (LDOS). We show that if the classical motion is chaotic, in a deep semi-classical region of a quantum system, both the shape of eigenstates and of the LDOS coincide with their classical analogs, on average. However, individual eigenstates exhibit quite large fluctuations which may not be treated as statistical ones. Thus, comparison of quantum quantities to the classical ones allows one to detect quantum effects of localization which for conservative systems emerge in the energy space.

DOI
10.1002/1521-3951(200007)220:1<721::aid-pssb721>3.0.co;2-0

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We study the model of two interacting particles moving in a 1D box, paying main attention to the quantum-classical correspondence for the average shape of quantum eigenstates and for the local density of states (LDOS). We show that if the classical motion is chaotic, in a deep semi-classical region of a quantum system, both the shape of eigenstates and of the LDOS coincide with their classical analogs, on average. However, individual eigenstates exhibit quite large fluctuations which may not be treated as statistical ones. Thus, comparison of quantum quantities to the classical ones allows one to detect quantum effects of localization which for conservative systems emerge in the energy space.
The Hamiltonian conservative system of two interacting particles has been considered both in classical and quantum description. The quantum model has been realized using a symmetrized two-particle basis reordered in the unperturbed energy. The main attention is paid to the structure of chaotic eigenfunctions (EF's) and to the local spectral density of states (LDOS). A remarkable correspondence has been found for the shapes of EF's and the LDOS in the energy representation to their classical counterparts. Comparison with the band random matrix theory predictions has revealed quite significant differences, which are due to the dynamical nature of the model. On the other hand, a partial agreement is found by inserting randomness ad hoc in the dynamical model for two-body matrix elements. This shows that, at least for small number of particles, care must be taken when classical correlations are neglected. The question of quantum localization in the energy space is discussed for both the dynamical and random models.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
Quantum–classical correspondence for the average shape of eigenfunctions and the local spectral density of states are well-known facts. In this paper, the fluctuations of the quantum wavefunctions around the classical value are discussed. A simple random matrix model leads to a Gaussian distribution of the amplitudes whose width is determined by the classical shape of the eigenfunction. To compare this prediction with numerical calculations in chaotic models of coupled quartic oscillators, we develop a rescaling method for the components. The expectations are broadly confirmed, but deviations due to scars are observed. This effect is much reduced when both Hamiltonians have chaotic dynamics.
It is generally believed that the classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences provides a deeper understanding of foundational aspects and has attracted a great deal of attention since the early days of quantum theory. It has been proposed that since a quantum mechanical wave function describes an intrinsic statistical behavior, its classical limit must correspond to a classical ensemble—not to an individual particle. This idea leads us to ask how the uncertainty product of canonical observables in the quantum realm compares with the corresponding dispersions in the classical realm. In this paper, we explore parallels between the uncertainty product of position and momentum in stationary states of quantum systems and the corresponding fluctuations of these observables in the associated classical ensemble. We confine ourselves to one-dimensional conservative systems and show, with the help of suitably defined dimensionless physical quantities, that first and second moments of the canonical observables match with each other in the classical and quantum descriptions—resulting in identical structures for the uncertainty relations in both realms.
The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.
We study the emergence of Boltzmann's law for the "single-particle energy distribution" in a closed system of interacting classical spins. It is shown that for a large number of particles Boltzmann's law may occur, even if the interaction is very strong. Specific attention is paid to classical analogs of the average shape of quantum eigenstates and "local density of states," which are very important in quantum chaology. Analytical predictions are then compared with numerical data.