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 single-particle 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 non-interacting spin-1
/
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 single-particle 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 Fermi-Dirac 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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- Winter '16
