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Journal of Russian Laser Research

On the relation between Schrödinger and von Neumann equations

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan, F. Zaccaria

Published Sep 1, 1999• 22 citations
Publisher
DOI
10.1007/bf02508909

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•Jan 1, 2010•0 citations
A generalized approach to the quantization of a large class of maps on a torus, i.e., quantization via the von Neumann equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well-behaved mixed quantum states for tori for which the corresponding Schr\"odinger equation has no solutions, as well as an extended spectrum for tori where the Schr\"odinger equation can be solved. Quantum-classical correspondence is demonstrated for the class of mappings considered, with the Wigner-Weyl density p(p,q,t) going to the correct classical limit. An application to the cat map yields, in a direct manner, nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the correspondence limit.
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.