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International Journal of Quantum Chemistry

Quantum phase‐space densities for a quartic oscillator

Gonzalo G. de Polavieja, F. Borondo, R. M. Benito

Published Sep 5, 1994• 9 citations
Publisher

Abstract

Abstract A quantum analog to the classical Poincaré surface of section, based on the Husimi function, is calculated for the quartic oscillator: H = 1/2(P + P ) + x2y2/2 + α(x4 + y4). The corresponding quantum densities in phase space are analyzed, and they show certain localization properties for a large number of states. © 1994 John Wiley & Sons, Inc.

DOI
10.1002/qua.560510620

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Abstract A quantum analog to the classical Poincaré surface of section, based on the Husimi function, is calculated for the quartic oscillator: H = 1/2(P + P ) + x2y2/2 + α(x4 + y4). The corresponding quantum densities in phase space are analyzed, and they show certain localization properties for a large number of states. © 1994 John Wiley & Sons, Inc.
Excited states of a model Hamiltonian with strongly coupled modes are investigated. Comparisons are made between the classical Poincaré surfaces of section and the Husimi function, which is a quantum mechanical phase-space density function. It is found that for the majority of states the Husimi function exhibits local maxima which resemble in shape and location the invariant tori of Poincaré surfaces of section. For certain anomalous states the maxima in the Husimi function bears no such resemblance, and these states are considered as quantum chaotic.
In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.
The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.