Eur. Phys. J. D 42, 153-162 (2007)
https://doi.org/10.1140/epjd/e2006-00278-8

Time evolution of the classical and quantum mechanical versions of diffusive anharmonic oscillator: an example of Lie algebraic techniques

Departamento Acadêmico de Disciplinas Básicas, Centro Federal de Educação Tecnológica de Minas Gerais, Belo Horizonte, MG, 30510-000, Brazil

Corresponding author: ^a jgpfaria@des.cefetmg.br

Received: 31 August 2006
Revised: 9 November 2006
Published online: 22 December 2006

Abstract

We present the general solutions for the classical and quantum dynamics of the anharmonic oscillator coupled to a purely diffusive environment. In both cases, these solutions are obtained by the application of the Baker-Campbell-Hausdorff (BCH) formulas to expand the evolution operator in an ordered product of exponentials. Moreover, we obtain an expression for the Wigner function in the quantum version of the problem. We observe that the role played by diffusion is to reduce or to attenuate the the characteristic quantum effects yielded by the nonlinearity, as the appearance of coherent superpositions of quantum states (Schrödinger cat states) and revivals.