Statistical Mechanics - Physics at Oregon State University

8.4. DENSITY-MATRIX APPROACH (BRAGG-WILLIAMS APPROXIMATION.171
eigenvectors e1(σ) and e2(σ). We have λ1 + λ2 = 1. The eigen equations are
now:
ρem = λjm [Tr ρ] em
(8.63)
The last result is interesting. It implies that the trace of the operator is
a simple factor, which does not alter the eigenstates, and does not affect the
ratio of probabilities for spin up and down. Also, the resulting factor in the
complete density matrix ρ is [Tr ρ] N . First of all, this trace is equal to one, and
hence the trace of ρ has to be a phase factor. Second, the phase factor is not
important, since it does not affect the total density matrix. Therefore, we can
take Tr ρ = 1.
We also require that the total density matrix ρ is positive definite, meaning
that
< Ψ|ρ|Ψ >< Ψ|ρ 2 |Ψ > (8.64)
for all states |Ψ >. Suppose the components of |Ψ > are given by
< σ1, · · · , σN|Ψ >=
en(i)(σi) (8.65)
where the function n(i) gives either one or two. We have
< Ψ|ρ|Ψ >= λ P 1 λ N−P
2
i
[Tr ρ] N = λ P 1 λ N−P
2
(8.66)
where P is the number of times n(i) is equal to one, and we used the fact that
Tr ρ = 1. Similarly, we have
As a result we have λ P 1 λ N−P
2
< Ψ|ρ 2 |Ψ >= λ 2P
1 λ 2N−2P
2 [Tr ρ] 2N = λ 2P
1 λ 2N−2P
2
λ 2P
1 λ 2N−2P
2
(8.67)
0 for all values of P . Taking
P = N gives λN 1 λ2N 1 which means (with λ1 being real) that λN 1. This
implies λ1 1, and since λ1 + λ2 = 1 we have λ2 0. Similarly, λ2 1 and
λ1 0. Therefore, ρ is positive definite with trace one.
The most general form of a matrix ρ is
1
ρ = 2 (1 + m) a∗
1
(8.68)
a 2 (1 − m)
Here m is again the average value of the spin variable, m =
σ σρ(σ, σ). Also,
the trace of this matrix is one. The number a is complex. Therefore, we are
left with three free parameters only! The number is three since a is complex
and a combination of two independent real numbers. Note that at this moment
m is still a parameter, but it will be the average magnetic moment after we
completed the minimization.
We started from the following minimization:
G(h, T ) = min
ρ Tr [ρH − ρhM + kBT ρ log(ρ)] (8.69)