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