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Journal of Mathematical Physics

Moyal equation—Wigner distribution functions for anharmonic oscillators

Trong Tuong Truong

Published Oct 1, 2021• 1 citation
Publisher

Abstract

We investigate the structure of Wigner distribution functions of energy eigenstates of quartic and sextic anharmonic oscillators. The corresponding Moyal equations are shown to be solvable, revealing new properties of Schrödinger eigenfunctions of these oscillators. In particular, they provide alternative approaches to their determination and corresponding eigenenergies, away from the conventional differential equation method, which usually relies on the not so widely known bi- and tri-confluent Heun functions.

DOI
10.1063/5.0021132

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We investigate the structure of Wigner distribution functions of energy eigenstates of quartic and sextic anharmonic oscillators. The corresponding Moyal equations are shown to be solvable, revealing new properties of Schrödinger eigenfunctions of these oscillators. In particular, they provide alternative approaches to their determination and corresponding eigenenergies, away from the conventional differential equation method, which usually relies on the not so widely known bi- and tri-confluent Heun functions.
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We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.
The formalism of generalized Wigner transformations developped in a previous paper, is applied to kinetic equations of the Lindblad type for quantum harmonic oscillator models. It is first applied to an oscillator coupled to an equilibrium chain of other oscillators having nearest-neighbour interactions. The kinetic equation is derived without using the so called rotating-wave approximation. Then it is shown that the classical limit of the corresponding phase-space equation is independent of the ordering of operators corresponding to the inverse of the generalized Wigner transformation, provided the latter is involutive. Moreover, this limit equation, which conserves the probabilistic nature of the distribution function and obeys an H-theorem, coincides with the kinetic equation for the corresponding classical system, which is derived independently and is distinct from that usually obtained in the litterature and not sharing the above properties. Finally the same formalism is applied to more general model equations used in quantum optics and it is shown that the above results remain unaltered.
New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.