Statistical Mechanics - Physics at Oregon State University

7.1. RELATION BETWEEN QUANTUM AND CLASSICAL MECHANICS.141
purpose to illustrate that a connection between quantum mechanical density
operators and classical distribution functions can be made. In general, the
expressions one needs for the distribution function W are more complicated
than the form given by Wigner.
Classical density matrix.
In order to give the formulas for classical statistical mechanics we consider
the canonical ensemble first. Since the Hamiltonian in the classical limit is
diagonal in X, we find that the density matrix is diagonal too, and that
< X|ρ| X >= ρ( X) = Ce −βH( X)
(7.7)
In many ways this form of ρ can be directly compared with the function W
introduced in the previous paragraph.
Volume per state.
The normalization constant C follows again from the requirement that Trρ =
1. This trace implies a sum over all possible states. In phase space, this is
replaced by an integral. We do have to multiply this integral by a volume factor,
though. The magnitude of this volume factor follows from the old quantum
mechanical quantization condition
pdq = nh (7.8)
where p is the generalized momentum corresponding to the generalized coordinate
q. This contour integral is an integral multiple of Planck’s constant h. In
two-dimensional phase space [p, q] the area inside a quantum orbit is nh, hence
the area between two quantum orbits is h itself. In other words, the area per
orbit is h. This value is consistent with the Heisenberg uncertainty relation.
If we construct a wave-package from plane-waves, we find that the position is
specified within ∆x and the corresponding momentum coordinate within ∆p.
Also, ∆x∆p > α for a value of α which is about one. If we represent the states
of a system by a grid of points in the p-x plane the distance between the points
in the x-direction has to be about ∆x. If grid points are closer together than
∆x they could not be distinguished and if they are further apart they do not
cover all space. Similarly, the distance of the grid points in p-space should be
∆p and hence the volume per point in one dimension should be ∆x∆p, or about
α. The formula above shows that we have to take α = 2π.
Classical integrals.
If the N particles in our system are not identical, we therefore have the
rule