Find
P
for fast rotation (Ω =
ω
) and its asymptotic form in the limit of ultrafast
rotation (Ω
ω
), and interpret your results.
2
Useful reminders:
The canonical partition function of a classical gas of
N
identical particles is
Z
=
1
h
2
N
N
!
(R
d
2
r
d
2
p
e

βh
(
r
,
p
)
)
N
, where
h
(
r
,
p
) is the singleparticle Hamiltonian.
R
R
0
re

λr
2
d
r
=
1

e

λR
2
2
λ
(recall the Jacobian in polar coordinates is d
x
d
y
=
r
d
r
d
θ
)
R
∞
∞
e

λu
2
d
u
=
p
π
λ
A
·
(
B
×
C
) =
B
·
(
C
×
A
) =
C
·
(
A
×
B
)
A
×
(
B
×
C
) = (
A
·
C
)
B

(
A
·
B
)
C
3
Problem 2: Statistical Mechanics II
Consider noninteracting spin1
/
2 fermions in two dimensions (2D) with a linear dis
persion relation
±
(
~
k
) =
±
~
v

~
k

.
Positive energy states (with energy
+
) define the
conduction
band and negative energy
states (with energy

) define the
valence
band.
Assume that the allowed wavevectors
~
k
=
{
k
x
, k
y
}
correspond to periodic boundary conditions over a square region of area
A
.
At temperature
T
= 0 the valence band is completely filled and the conduction band is
completely empty. At finite
T
, excitations above this ground state correspond to adding
particles
(occupied states) in the conduction band or
holes
(unoccupied states) in the
valence band.
(a) (25 points) Find the singleparticle density of states
D
( ) as a function of the energy
in terms of
~
, v, A
. Sketch
D
( ) over both the negative and positive energy region.
In the next two parts we will argue that the chemical potential
μ
(
T
) = 0 at any
temperature
T
(which can be assumed so for the rest of the problem).
(b) (20 points) Using the FermiDirac distribution, show that if
μ
(
T
) = 0 then the prob
ability of finding a particle at energy
is equal to the probability of finding a hole at
energy

.
(c) (15 points) Particle number conservation requires that the number of particles in the
conduction band must equal the number of holes in the valence band,
N
p
=
N
h
. Show
that if
μ
(
T
) = 0 then this condition holds at all
T
. (Do not worry if the integrals are
formally divergent – they will be cut off in any physical system.)
(d) (25 points) Find the total internal energy of the excitations above the
T
= 0 state,
U
(
T
)

U
(0), expressed in terms of
A, v,
~
, k
B
. Note that since we are subtracting
U
(0),
in the valence band you only need to count the energy associated with holes.
Useful integral:
1
n
!
Z
∞
0
x
n
e
x
+ 1
= (1

2

n
)
ζ
(
n
+ 1)
,
where
ζ
(
n
+ 1) =
∑
∞
k
=1
k

n

1
is the Riemann zeta function.
(e) (15 points) Use your answer in (d) to find the heat capacity at constant area
C
A
(
T
)
∝
T
α
. What is the exponent
α
?
4
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