Statistical Mechanics - Physics at Oregon State University

A.4. SOLUTIONS FOR CHAPTER 4. 247
Problem 9
1 − k Z(T, µ) = 1 + e B T (ɛ1−µ) 1 − k + e B T (ɛ2−µ) 1 − k + e B T (ɛ1+ɛ2+I−2µ)
N(T, µ) = 1
1 − k e B T
Z
(ɛ1−µ) −
+ e 1
kB T (ɛ2−µ) −
+ 2e 1
kB T (ɛ1+ɛ2+I−2µ)
∂N
= −
∂T µ
1
Z2
∂Z
1 − k e B T
∂T µ
(ɛ1−µ) 1 − k + e B T (ɛ2−µ) 1 − k + 2e B T (ɛ1+ɛ2+I−2µ)
+
1
1 − k (ɛ1 − µ)e B T
Z
(ɛ1−µ) 1 − k + (ɛ2 − µ)e B T (ɛ2−µ)
1 − k + 2(ɛ1 + ɛ2 + I − 2µ)e B T (ɛ1+ɛ2+I−2µ)
ZkBT 2
∂N
=
∂T µ
1
kBT 2
1 − k −N(T, µ) (ɛ1 − µ)e B T (ɛ1−µ) 1 − k + (ɛ2 − µ)e B T (ɛ2−µ)
1 − k + (ɛ1 + ɛ2 + I − 2µ)e B T (ɛ1+ɛ2+I−2µ)
+
1 − k (ɛ1 − µ)e B T (ɛ1−µ) 1 − k + (ɛ2 − µ)e B T (ɛ2−µ)
1 − k + 2(ɛ1 + ɛ2 + I − 2µ)e B T (ɛ1+ɛ2+I−2µ)
=
1 − k (1 − N(T, µ))(ɛ1 − µ)e B T (ɛ1−µ) 1 − k + (1 − N(T, µ))(ɛ2 − µ)e B T (ɛ2−µ) +
1 − k (2 − N(T, µ))(ɛ1 + ɛ2 + I − 2µ)e B T (ɛ1+ɛ2+I−2µ)
This can be negative if N(T, µ) > 1, which requires that the lowest energy state
is ɛ1 + ɛ2 + I, or I < −ɛ1 − ɛ2. In that case the binding energy between the
two particles (−I) is so strong that particles are most likely to be found two at
a time. In that case N(T = 0, µ) = 2 and this value can only become smaller
with temperature.
A.4 Solutions for chapter 4.
Problem 4.
The Maxwell distribution function fM is given by fM (ɛ; T, µ) = e 1
k B T (µ−ɛ) .
Show that