Abstract
We show that the non-linear semi-quantum Hamiltonians which may be expressed as(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:(whereis the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.
Highlights
There exists certain kind of semiquantum non-linear systems which can be represented by the following Hamiltonian [1]-[3]How to cite this paper: Sarris, C.M. and Plastino, A. (2014) Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras
{ } Oj (t ) = Tr ρ (t )Oj and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values
( ) quantum Hamiltonian given by Equation (2), H = Tr ρ H, provided that it is a linear superposition of the generators of some Lie algebra, 3) the mean value of the semiquantum Hamiltonian, His taken to coincide with a Hamiltonian function [2] [3] [5] [6] that, in turn, generates the temporal evolution of the classical degrees of freedom q and p, 4) since some Lie algebra has been associated to the semiquantum system, it will be possible to derive some dynamic invariants which, in turn, can be expressed in terms of the mean val
Summary
There exists certain kind of semiquantum non-linear systems which can be represented by the following Hamiltonian [1]-[3]. ( ) quantum Hamiltonian given by Equation (2), H = Tr ρ H , provided that it is a linear superposition of the generators of some Lie algebra, 3) the mean value of the semiquantum Hamiltonian, His taken to coincide with a Hamiltonian function [2] [3] [5] [6] that, in turn, generates the temporal evolution of the classical degrees of freedom q and p , 4) since some Lie algebra has been associated to the semiquantum system, it will be possible to derive some dynamic invariants which, in turn, can be expressed in terms of the mean val-.
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