Statistical Mechanics - Physics at Oregon State University

A.6. SOLUTIONS FOR CHAPTER 6. 259
for any variable X. Using R, R∗ , and x for X and setting the result to zero
gives:
0
0 = T r
0
1
0
0
log(ρ) + T r
0
1
0
0
0 = T r
1
0
0
0
log(ρ) + T r
1
0
0
1
0 = T r
0
0
1
log(ρ) + T r
−1
0
0
−1
or
Therefore
and hence
(log(ρ)) 21 = 0
(log(ρ)) 12 = 0
(log(ρ)) 11 − (log(ρ)) 22 = 0
log(ρ) =
ρ =
y 0
0 y
z 0
0 z
Using the fact that the trace is one we finally get:
1
ρ = 2 0
0 1
2
or R = 0,x = 1
2 .
The same can be obtained by finding the eigenvalues of ρ.
The second order derivatives are all negative or zero. Use result (9,33) for any
of the elements and change to coordinates R, R ∗ , and x.
Problem 6.
A quantum mechanical system is described by a simple Hamiltonian H, which
obeys H 2 = 1. Evaluate the partition function for this system. Calculate the
internal energy for T → 0 and T → ∞.
Z = T r
∞
n=0
1
n! (−1)n β n H n