13 - Quantum Hamiltonians and Kinetic Equations

Abstract

This chapter investigates the spectrum of Hamiltonians in application to quantum optics, outlining several details related to quantum Hamiltonians and “classical” kinetic equations. It uses the same “ideology” applying conservation laws like earlier, studying discrete models of Boltzmann equation. This approach to the conservation laws reduced the dimension of the spectrum problem to the finite-dimensional one and it was already used by physicists in construction of frequency convertors. It represents efforts in revealing the relations between different type of mathematical equations and modern interpretation of mathematical physics. Conservation laws linear by particles densities play important role in the theory of kinetic equations. In the case of Boltzmann equation, they are fundamental macroscopic values necessary for introduction of continuous medium, when the hydrodynamics equations for mean values of density, impulse, and energy are written. In the homogeneous space case of the Boltzmann equation, these laws completely define the qualitative behavior of the system. H-theorem justifies tending of the system to stationary distribution whose parameters are defined from the corresponding conservation laws. For the classical Boltzmann equation, this distribution is called the Maxwellian. The idea of this correspondence allows to write down the generalizations of discrete models for Boltzmann equations based on conservation laws for quantum Hamiltonians (QH) and kinetic equations (KE) either for cases of annihilation and particle creation or triple (and even higher) order collisions. This generalization refers to a class of equations in chemical kinetics where the H-theorem holds.