Statistical Mechanics - Physics at Oregon State University

Chapter 7
Classical statistical
mechanics.
7.1 Relation between quantum and classical mechanics.
Choice of basis.
Based on the knowledge of quantum statistical mechanics, it is straightforward
to derive the expressions valid for a classical description of a many particle
system. A classical state of a system is defined by the values of the positions
ri and momenta pi. These vectors are combined in a single vector X in a 6Ndimensional
phase-space. Every value of X corresponds to a certain state of
the system. The classical Hamiltonian H( X) is constructed from the quantum
mechanical one by replacing momentum operators by numbers pi. A good basis
for the expressions for the density matrix is therefore the set of all vectors | X >.
In quantum mechanics it is impossible to specify both the position and the
momentum of a particle at the same time, since the position and momentum
operators do not commute. The commutators are proportional to and one
way of thinking about the classical limit of quantum mechanics is to assume
that → 0. In this sense the position and momentum operators do commute
in the classical limit and we are allowed to use their simultaneous eigenvalues
as labels characterizing the state of a system.
Wigner distribution function.
In a more formal approach we use so-called Wigner distribution functions,
defined by
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