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On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation

Fedele R., S. DE NICOLA, Jovanovic D, D. Grecu, Visinescu A

Published Jan 1, 2010• 0 citations

Abstract

The Wigner transformation is used to define the quasidistribution(Wigner function) associated with the wave function of the cylindrical nonlinear Schrodinger equation (CNLSE) in a way similar to that of the standard nonlinear Schr¨odinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the cylindrical nonlinear von Neumann equation (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the cylindrical Wigner function and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically andevaluated numerically.

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The Wigner transformation is used to define the quasidistribution(Wigner function) associated with the wave function of the cylindrical nonlinear Schrodinger equation (CNLSE) in a way similar to that of the standard nonlinear Schr¨odinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the cylindrical nonlinear von Neumann equation (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the cylindrical Wigner function and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically andevaluated numerically.

We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrödinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.

We discuss a family of quasidistributions (

s

-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr\"odinger equation. It turns out that although Wigner functions satisfy the Schr\"odinger equation in phase space, they have a completely different interpretation.

The Wigner function, a kind of wavelet transform is studied for one- and two-soliton solutions to the nonlinear Schrödinger equation. The exact expression of the Wigner transform for the one-soliton solution is obtained. The features of soliton-soliton interaction of different two-soliton states, the states of coherence analogous to that of two-localized wave packets in quantum mechanics, are displayed. Suitability of this transform for analyzing “ soliton ” in open systems is also discussed.

A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function

S^{(K)}(q,p,t)

whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing

\frac{\partial }{\partial p}

by 0 and p by

\frac{\hbar }{\imath } \frac{\partial }{\partial q}

, thus providing an explanation for the common quantization rules.