Semiclassical model for quantum dissipation

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

“…The motion invariant given by Equation (38) is the so-called second order centered invariant of [29] [31] and its value can be fixed through the initial conditions imposed on the system by means of Equations (20) and (21). Our interest in the particular case given by Equation (38) lies in the fact that it is possible to demonstrate that Equation (38) represents a positive definite quadratic form [32].…”

“…which is the generalized Ehrenfest theorem given by Equation (21). In short, if we are able to close a semi Lie algebra under commutation with the semiquantum non-linear Hamiltonian (16), then we can integrate the equations of motion of the quantum degrees of freedom even though the Hamiltonian exhibits a nonlinearity via the ( ), j j a q p O [23].…”

“…Let us consider a mixed physical system represented by the semiquantum Hamiltonian represented by Equation (15) with a coupling term [1] [12] [20] [24]- [26]. As the classical degrees of freedom act like as if they were parameters, the Hamiltonian given by Equation (15) may be cast as [3] [23] ( ) ( ) freedom.…”

“…These motion invariants always may be expressed in terms of the quantum degrees of freedom's mean values given by Equation (23). Concerning the system's classical degrees of freedom, the energy is taken to coincide with the quantum expectation value of the semiquantum Hamiltonian [1] [2] [6] [17] [21] given by Equation (29) and the temporal evolution of the classical variables is given by [2] …”

“…There exists certain kind of semiquantum non-linear systems which can be represented by the following Hamiltonian [1]- [3] …”