Hamiltonian operators statistical mechanics

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

In Section 2.5.3 we derived the semiclassic expression for the ceinonical partition function [see Eq. (2.110)] based on the assumption that at sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltonian function [see Elq. (2.100)]. In this section we will sketch a more refined treatment of the semiclassic theory developed in Section 2.5 originally due to Hill and presented in detail in his classical work on stati.stical mechanics [326]. Because of Hill s clear and detailed exposition and because we need the final result mainly as a justification to treat confined fluids by means of classic statistical thermodynamics, we will just briefly outhne the key ideas of Hill s treatment for reasons of completeness of the current work. 

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

This contribution deals with the description of molecular systems electronically excited by light or by collisions, in terms of the statistical density operator. The advantage of using the density operator instead of the more usual wavefunction is that with the former it is possible to develop a consistent treatment of a many-atom system in contact with a medium (or bath), and of its dissipative dynamics. A fully classical calculation is usually suitable for a many-atom system in its ground electronic state, but is not acceptable when the system gets electronically excited, so that a quantum treatment must then be introduced initially. The quantum mechanical density operator (DOp) satisfies the Liouville-von Neumann (L-vN) equation [1-3], which involves the Hamiltonian operator of the whole system. When the system of interest, or object, is only part of the whole, the treatment can be based on the reduced density operator (RDOp) of the object, which satisfies a modified L-vN equation including dissipative rates [4-7]. 

Traditional wisdom in the field of nonequilibrium statistical mechanics has it that one gets rid of recurrences, in the thermodynamic limit, because the spectrum of the Hamiltonian becomes dense or continuous. This kind of statement must be taken cum grano salts. First, it is next to impossible to exhibit, in the usual formalism, an operator that would be mathematically well defined and that would be obtained from a finite system Hamiltonian by the limiting procedures that Hilbert space techniques (as commonly used) have to offer. Second, if this problem were to be ignored for a while, it would still be of interest to know how dense or continuous things become in the thermodynamic limit. And third, it would be useful to link these how much questions and answers with the rate of approach to equilibrium. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

As soon as large ensembles of particles with statistical populations of the eigenstates and incoherent exchange and relaxation processes between these states are investigated, quantum statistical tools are necessary to describe the system. In this situation the quantum mechanical density operator p has to be employed. For the coherent evolution of the density operator under the influence of a Hamiltonian H, the following differential equation is found [80]... 

---