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Physical Review A

Comment on ‘‘Information theory and nonlinear quantal Hamiltonians’’

Francisco Marcelo Fernández

Published Jun 1, 1990• 2 citations
Publisher

Abstract

A recently discussed Hamiltonian approach to dissipation effects is considered that leads to nonlinear equations for the time evolution of the expectation values of some operators. It is shown here that the Heisenberg equations of motion for the coordinate and momentum operators can be exactly solved, allowing a better study of the model.

DOI
10.1103/physreva.41.7066

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A recently discussed Hamiltonian approach to dissipation effects is considered that leads to nonlinear equations for the time evolution of the expectation values of some operators. It is shown here that the Heisenberg equations of motion for the coordinate and momentum operators can be exactly solved, allowing a better study of the model.
We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.
Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.
Quantum systems interacting with a dissipative environment can effectively be described by a nonlinear modification of the Schrodinger equation. For some standard problems that possess analytical solutions, e.g. the free motion and the harmonic oscillator, also the nonlinear equation can be solved in a closed form and has Gaussian-shaped wave packet solutions. The equation determining the time-evolution of the wave-packet width is - already in the case of the usual conservative Schrodinger equation - nonlinear and provides the quantities that are necessary to construct the Green’s function for the respective system. The Green’s functions for the conservative and dissipative cases will be given explicitly in position and momentum space.