The Characteristic Times of Violation of Quasiclassical Approximation for Integrable Boson and Spin Systems

Abstract

AbstractIn the previous sections we were mainly interested in considering the methods which can allow us to analyse the quantum corrections to the classical solutions for different boson and spin systems. As a first step we consider in this section the application of the considered above c-number equations for quantum integrable boson and spin systems in quasiclassical region of parameters. We derive one of the main characteristic dynamical parameter, namely the characteristic time-scale τ ħ of violation of the quasiclassical solutions. For this we introduce below in section 5.1 a function δ(t) which characterises the difference in time between quantum and corresponding classical solutions. It will be shown that in general case the universal time-scale τ ħ which depends only on the parameters of the Hamiltonian does not exist. Namely, the time-scale τ ħ depends also on the expectation value one measures, and, for example, for big momentums this time-scale τ ħ can significantly decrease. Nevertheless, for the typical cases of not big momentums there exists rather universal time-scale τ ħ which can be expressed in the terms of two mentioned above main parameters - classical parameter of nonlinearity \( \overline \mu \) and a quasiclassical parameter κ = I/ħ = 1/ε (where I is a characteristic action of the system): \( \tau \hbar \sim \sqrt \kappa /\overline \mu \). In section 5.2 we investigate the problem of time-scale τ ħ taking into account an additional averaging over the initial density matrix.KeywordsClassical LimitClassical ParameterDiagonal Matrix ElementBoson SystemAdditional AverageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.