arXiv:nlin/0201005 · Journal of Physics A Mathematical and General

Semiclassical analysis of Wigner functions

Gregor Veble, Marko Robnik, Valery G. Romanovski

Published Apr 26, 2002• 1 citation

Abstract

In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.

DOI: 10.1088/0305-4470/35/18/311

Related papers

ON THE PROPAGATION OF SEMICLASSICAL WIGNER FUNCTIONS

P. P. De, Pedro de M. Rios, A. M. Ozorio +1 more

Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian

Jan 1, 200139 citations

Semiclassical Analysis of Wave Dynamics via Wigner Transforms: Applications to the Quantum Harmonic, Anharmonic, and Dirac Oscillator

August Burg

In this Master's thesis, we give a pedagogical introduction to the semiclassical analysis of wave dynamics in single-band and multi-band systems. We show how the semiclassical evolutions of Wigner functions can approximate the dynamics of wave packet centers and wave packet deformations. Three systems of increasing complexity are considered: the quantum harmonic oscillator and the quantum anharmonic oscillator as single-band systems, and the Dirac oscillator as a two-band system. We explain the origins of the first-order semiclassical corrections (Moyal, Littlejohn-Flynn, and Berry) in the equations of motion and examine their effects on predictive accuracy. Another focus is set on the conditions for validity of the semiclassical analysis. For both the quantum anharmonic and the Dirac oscillators, a dimensionless parameter controlling the accuracy of the approximation is identified. In multi-band systems, the formalism applies only to states within a single adiabatic subspace, which, for the Dirac oscillator, corresponds to an energy branch. This motivates the construction of single-branch almost coherent states: approximately Gaussian wave packets formed by superpositions of states from a single energy branch. Their centers undergo oscillations with an aperiodic envelope. These states enable comparison between semiclassical Wigner function evolution and the commonly used Sundaram-Niu formalism. We show that semiclassical Wigner function evolution captures deformation-related dynamics, including the initial decay of oscillations, which the Sundaram-Niu formalism cannot describe by construction. Additionally, exact quantum evolution reveals partial relocalization of these single-branch almost coherent states, a phenomenon that cannot be captured by the semiclassical evolution.

cond-mat.mes-hallCross-listedarXiv:2610.22414Oct 16, 20260 citations

On the Group of Translations and Inversions of Phase Space and the Wigner Functions

Jens Peder Dahl

Grossmann and Royer have recently shown that the Wigner functions are closely related to the set of all translations and inversions of phase space. This allows the phase space representation of quantum mechanics to be constructed directly on the group of phase space translations and inversions. Starting from this observation, we have derived analytical expressions for the matrix elements of the translation and inversion operators, in the harmonic oscillator representation, without introducing coordinate or momentum wavefunctions.

Physica ScriptaApr 1, 198246 citations

Quantum and semiclassical phase functions for the quantization of symmetric oscillators

A. Matzkin, M. Lombardi

We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville–Green transformations of the Schrödinger equation. For anharmonic oscillators, first-order semiclassical quantization is seldom accurate and the higher order expansions eventually break down given the asymptotic nature of the series. A quantum phase that allows us in principle to retrieve the exact quantum mechanical quantization condition and wavefunctions is given along with an iterative scheme to compute it. The arbitrariness surrounding quantum phase functions is lifted by supplementing the phase with boundary conditions involving high order semiclassical expansions. This allows us to extend the definition of oscillation numbers, which determine the quantization of the harmonic oscillator, to the anharmonic case. Several illustrations involving homogeneous as well as coupling constant dependant anharmonic oscillators are given.

arXiv:0706.2978Journal of Physics A Mathematical and GeneralJun 22, 20051 citation

On the propagation of semiclassical Wigner functions

P P de M Rios, A M Ozorio de Almeida

We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we rediscuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centred on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.

arXiv:math-ph/0111012Journal of Physics A Mathematical and GeneralMar 8, 200240 citations

Wigner Function for Harmonic Oscillator and The Classical Limit

Jan Mostowski, Joanna Pietraszewicz

The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.

arXiv:2104.06638Apr 14, 20211 citation