Page 41 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

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Quantum mechanical density matrix

For systems which obey quantum mechanics, the formulation of the problem of
treating large numbers of particles is, of course, somewhat different than it is for
classical systems. The microscopic description of the system is provided by a wave
function which (in the absence of spin) is a function of the classical coordinates
{q i }. The mathematical model is provided by a Hamiltonian operator H which is
often obtained from the corresponding classical Hamiltonian by the replacement
p i → (¯h/i)(∂/∂q i ). In other cases the form of the Hamiltonian operator is simply
postulated. The microscopic dynamics are provided by the Schr¨odinger equation
i¯h(∂ /∂t) = H which requires as initial condition the knowledge of the wave
function ({q i }, t) at some initial time t 0 . (Boundary conditions on ({q i }, t) must
be specified as part of the description of the model as well.) The results of ex-
periments in quantum mechanics are characterized by operators, usually obtained,
like the Hamiltonian, from their classical forms and termed observables. Operators
associated with observables must be Hermitian. In general, the various operators
corresponding to observables do not commute with one another. It is possible to
find sets of commuting operators whose mutual eigenstates span the Hilbert space
in which the wave function is confined by the Schr¨odinger equation and the bound-
ary conditions. A set of such (time independent) eigenstates, termed ψ ν (q), is a
basis for the Hilbert space. The relation between operators φ op and experiments is
provided by the assumed relation

¯ ∗ 3N
φ(t) = ({q i }, t)φ op ({q i }, t)d q (2.1)
¯
where φ(t), the quantum mechanical average, is the average value of the experi-
mental observable associated with the operator φ op which is observed on repeated
experimental trials on a system with the same wave function at the same time t.
Unlike the classical case, even before we go to time averages or to large systems,
only averages of the observed values of experimental variables are predicted by the

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