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Elsevier eBooks

Quantum Hamiltonians and Kinetic Equations

Victor Valentinovich Vedenyapin, Alexander V. Sinitsyn, Eugene Dulov

Published Jan 1, 2011• 0 citations
Publisher
DOI
10.1016/b978-0-12-387779-6.00013-2

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Abstract Based on a model open system, the kinetic equation obtained from the initial exact Hamiltonian with anti-rotating terms is shown to differ from the kinetic equation from the approximate effective Hamiltonian without anti-rotating terms. The paper proves necessity of transition from the exact Hamiltonian to the effective Hamiltonian, with the main idea of its construction being the requirement for the absence of rapidly varying terms in the interaction representation. This requirement is necessary to obtain the correct kinetic equation.
In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr\"odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.
Abstract We show that a number of particle decay problems, which are described by the non-Hermitian Hamiltonian (correspondingly, the nonunitary dynamics is spoken of), can be correctly and consistently reformulated in terms of stochastic differential equations upon application of the algebraic perturbation theory. In this formulation, kinetic equations are obtained in the standard scheme of the theory of open quantum systems. The parameters of the kinetic equation coincide with similar parameters describing the decay of a particle when the decay is considered on the basis of the non-Hermitian Hamiltonian.