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Journal of Physics A Mathematical and General

On the zeros of the Husimi distribution

H. J. Korsch, Christopher Müller, H Wiescher

Published Oct 21, 1997• 72 citations
Publisher

Abstract

The basic features of the zeros of the Husimi phase-space density of a quantum eigenstate are discussed. It is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials, as illustrated by numerical examples for the Morse oscillator. Also discussed is the distribution of zeros in quantum Poincaré sections for systems with two degrees of freedom.

DOI
10.1088/0305-4470/30/20/003

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International Journal of Quantum Chemistry•Sep 5, 1994•9 citations
The basic features of the zeros of the Husimi phase-space density of a quantum eigenstate are discussed. It is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials, as illustrated by numerical examples for the Morse oscillator. Also discussed is the distribution of zeros in quantum Poincaré sections for systems with two degrees of freedom.
We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.
We discuss some basic tools for an analysis of one-dimensionalquantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros.Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.
For the representation of eigenstates on a Poincare section at the boundary of a billiard different variants have been proposed. We compare these Poincare Husimi functions, discuss their properties, and based on this select one particularly suited definition. For the mean behavior of these Poincare Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincare Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincare Husimi functions in the case of ergodic systems is shown.
We propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. In the present article, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. There are other ways to obtain it, but we do not consider here. We specially discuss the problem of the continuous harmonic oscillator.