Statistical Mechanics - Physics at Oregon State University

128 CHAPTER 6. DENSITY MATRIX FORMALISM.
At this point we encounter an important difference between functions and
operators. We cannot interchange the order of operators if they do not commute,
and hence we have to write something of the form A 2 ′ = A ′ A+AA ′ ! In general,
an operator and its derivative do not commute. Therefore:
and
∂
∂x τ n (x) =
n
τ m−1
∂τ
(x) τ
∂x
n−m (x) (6.46)
m=1
∂
∂x eτ(x) = 1
n!
n
n
τ m−1
∂τ
(x) τ
∂x
n−m (x) (6.47)
m=1
and we cannot say much more. But, in our example we need to take the trace
of this equation, and we can use Tr (ABC) = Tr (CAB). Therefore:
Tr ( ∂
∂x eτ(x) ) = 1
n!
which finally leads to
or in other words
n
n
Tr (τ m−1
∂τ
(x) τ
∂x
n−m (x)) (6.48)
m=1
Tr ( ∂
∂x eτ(x) ) = 1
n!
n
Tr ( ∂
∂x eτ(x) ) =
Tr ( ∂
∂x eτ(x) ) = Tr (
n
n
n
Tr (τ n−1
∂τ
(x) ) (6.49)
∂x
m=1
1
n! nTr (τ n−1 (x)
1
(n − 1)! τ n−1 (x)
Tr ( ∂
∂x eτ(x) ) = Tr (e τ(x)
Tr ( ∂ρ
) = Tr (ρ
∂x
and using x = Rnm:
∂ log(ρ)
Tr (ρ
) = Tr ( ∂ρ
∂Rnm
∂ log(ρ)
∂Rnm
∂x
∂τ
) (6.50)
∂x
∂τ
) (6.51)
∂x
∂τ
) (6.52)
∂x
) (6.53)
∂Tr ρ
) = ( ) = δnm
∂Rnm
Our final result for the partial derivative is therefore
∂X
= −kB 〈m| log(ρ) |n〉 + (λ − 1)kBδnm
∂Rnm
and setting this equal to zero gives
(6.54)
(6.55)