Homework 6 Solution
Yuan Wan
1
Solution to Kardar Problem 4.10
(a)(5 points) We notice that the Hamiltonian of the system is written as the
sum of two commuting operators:
N
p2
i
B
2m
H = Horbital + Hinternal =
i
N
z
Si
[Horbital , Hinternal ] = 0 (1)
i
Homework 9 Solution
Yuan Wan
1
Part A
We introduce the following new elds:
= sin cos ;
= sin sin ;
We assume that both and are close to 0, and hence ,
above expressions and nd:
cos 1
and
(1)
1. We invert the
2 + 2
2
(2)
= 2
+ 2
(3)
( )2 + sin2 ( )2
Homework 8 Solution
Yuan Wan
1
Solution to Kardar Problem 5.8
(a)(5 points)
The Hamiltonian is written as
1
p2
i
+
2m 2
H=
i
V (ri rj )
(1)
i,j
The partition function is written as
Z = Z0 Z1
(2)
where Z0 is the free part:
d2 pi
p2
exp( i ) =
2
h
2m
Z0 =
Homework 7 Solution
Yuan Wan
1
Solution to Kardar Problem 5.10
(a)(20 points)
The partition function for N particles is given by:
1N
Z
N! 1
(1)
p2
d3 pd3 q
exp
+ eU (q)
h
2m
(2)
ZN =
where
Z1 =
Note the electrons carrry negative charge e. Integrating ou
Homework 10 Solution
Yuan Wan
1
Solution to Kardar 7.5
Part (a)
The joint probability is given by the following:
P [n(k)] =
1
Z
exp [ ( (k) )n(k)]
(1)
k
where Z is the partition function
Part (b) On the one hand, the average occupation number at momentum
Homework 11 Solution
Yuan Wan
1
Solution to Kardar 7.15
Part (a)
The joint probablity is given by
P [n(k)] =
1
Z
z (k)n(k)
(1)
k
Part (b) The above joint probablity is from products of independent probablity density:
P [n(k)] =
p(n(k)
(2)
k
where
p(n) = (
Homework 5 Solution
Yuan Wan
1
Solution to Problem 1
(a)(15 points)
We prove the following
P
N
B=N
(1)
T ,V
in part (a). We notice that the pressure P is an intensive property and thus a
0th order homogeneous function of V and N :
P (T, V, N ) = P (T, V,
Homework 4 Solution
Yuan Wan
1
Solution to Kardar 3.12
(a)(8 points)
The one-particle density should take the following form:
feq (p, q) = A exp(
p2
)
2mkB T
(1)
where A is the normalization constant. The normalization condition is
d3 pd3 qfeq (p, q) = N
Homework 3 Solution
Yuan Wan
1
Solution to Kardar 2.10
The distribution of the voltage is given by
p(V )dV =
1
2 2
eV
2
/2 2
dV
(1)
Hence, the distribution of the current is given by
p(V )dV = p[V (I )]
dV
dI = q (I )dI
dI
(2)
that is
dV
dI
By inverting