Statistical Mechanics - Physics at Oregon State University

140 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
W (
k, r) =
d 3 xe ı kx < r + 1
2
1
x|ρ|r − x > (7.1)
2
for a system with only one particle. In the formula above ρ is the density matrix
describing this system. This formula represents a partial Fourier transform. If
we want to transform the complete density matrix we need two exponentials
and two three dimensional integrations, because we need to transform the information
contained via r and r ′ in < r ′ |ρ|r > both. But now we only transform
half of the space, which still leaves us with a function of two three dimensional
vectors.
and
It is easy to show that
1
(2π) 3
1
(2π) 3
d 3 kW ( k, r) =< r|ρ|r > (7.2)
d 3 rW ( k, r) =< k|ρ| k > (7.3)
These are the probability functions for finding the particle at position r or
with momentum k. In order to derive the last expression we have used
3 −
|x >= (2π) 2 d 3 ke ıkx | k > (7.4)
to get
W (
k, r) =
d 3 xe ı kx
d 3 k ′
1
(2π) 3
d 3 k ′
d 3 k”e i[ k”(r− 1
2 x)− k ′ (r+ 1
2 x)] < k ′ |ρ| k” >=
d 3 k”δ( k − 1
2 [ k ′ + k”])e i[ k”r− k ′ r] < k ′ |ρ| k” > (7.5)
Integration over r gives an extra delta-function δ( k ′ − k”) and the stated result
follows.
Averages.
Suppose O( k, r) is a classical operator in phase space and if O is the quantummechanical
generalization of this classical operator. In a number of cases it is
possible to show that the quantum-mechanical expectation value of the operator
< O > is given by
< O >= d 3
k d 3 xO( k, x)W ( k, x) (7.6)
which relates classical values and quantum mechanical values of O and O. This
procedure is not always possible, though, and our discussion only serves the