Quantum dynamics basic properties

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

The basic properties of the classical Liouville equation and the troublesome questions they raise are shared by the quantum Liouville equation. For the quantum case, we smmnarize these properties as follows (i) The canonical density operator is stationary with respect to the quantum Liouville equation, (ii) The quantum Liouville operator t is Hemtitiaa (iii) The quantum Liouville equation is time-reversal invariant, (iv) The Gibbs entropy S t) is time independent when S t) is determined using the formal solution p(t) of the quantum Liouville equatioa Given the density operator p(t), we can determine the average value d t)) of the quantum dynamical variable 0 at time t by using the relation... 

Mary Jo Nye enumerates the topics treated in the Journal as molecular spectroscopy and molecular structures, the quantum mechanical treatment of electronic structure of molecules and crystals and the problem of chemical binding, the kinetics of chemical reactions from the standpoint of basic physical principles, the thermodynamic properties of substances and calculation by statistical mechanical methods, the structure of crystals, and surface phenomena. M.J. Nye, From Chemical Philosophy to Theoretical Chemistry. Dynamics of Matter and Dynamics of Disciplines (Berkeley University of California Press 1993), 254. Many of these were considered by Barriol, as we will see later in this chapter. 

In order to investigate the energies and molecular properties of molecules interacting with aerosol particles, it is crucial to establish the Hamiltonians and the energy functionals for the two structural environment methods. The basic principle for both structural environment methods is the same and it is one that has been utilized successfully within quantum chemistry [2-33] and molecular reaction dynamics [19,68-71,96], we divide a large system into two subsystems. The focus is... 

The introduction of statistical features in the basic molecular models is considered in Section II,E,2. It is argued that, in most cases, at least one part of the nonradiant molecular manifold is unknown and should be treated under statistical assumptions. By means of a partially random representation of the zero-order Hamiltonian, of the kind introduced by Wigner and others in statistical nuclear theory (see Bloch, 1966, 1969), we define a general dynamical model in which both the quantal and statistical properties of the molecular excitations are combined. Special attention is given to the nature of the statistical limit and irreversible radiationless transitions for the molecular excited states. We also discuss the relationship between this concept and similar concepts in quantum statistical theory of relaxation and master equations (Zwanzig, 1961). 

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