3Quantum gases
II Statistical Physics
3.1 Density of states
Consider an ideal gas in a cubic box with side lengths
L
. So we have
V
=
L
3
.
Since there are no interactions, the wavefunction for multiple states can be
obtained from the wavefunction of single particle states,
ψ(x) =
1
√
V
e
ik·x
.
We impose periodic boundary conditions, so the wavevectors are quantized by
k
i
=
2πn
i
L
,
with n
i
∈ Z.
The one-particle energy is given by
E
n
=
~
2
k
2
2m
=
4π
2
~
2
2mL
2
(n
2
1
+ n
2
2
+ n
2
3
).
So if we are interested in a one-particle partition function, we have
Z
1
=
X
n
e
−βE
n
.
We now note that
βE
n
∼
λ
2
L
2
n
2
,
where
λ =
r
2π~
2
mkT
.
is the de Broglie wavelength.
This
λ
2
L
2
gives the spacing between the energy levels. But we know that
λ L
. So the energy levels are very finely spaced, and we can replace the sum
by an integral. Thus we can write
X
n
≈
Z
d
3
n ≈
V
(2π)
3
Z
d
3
k ≈
4πV
(2π)
3
Z
∞
0
d|k| |k|
2
,
where in the last step, we replace with spherical polars and integrated over
angles. The final step is to replace |k| by E. We know that
E =
~
2
|k|
2
2m
.
So we get that
dE =
~
2
|k|
m
d|k|.
Therefore we can write
X
n
=
4πV
(2π)
3
Z
∞
0
dE
r
2mE
~
2
m
~
2
.
We then set
g(E) =
V
2π
2
2m
~
2
3/2
E
1/2
.
This is called the density of states. Approximately,
g
(
E
) d
E
is the number of
single particle states with energy between E and E + dE. Then we have
X
n
=
Z
g(E) dE.
The same result (and derivation) holds if the sides of the box have different
length. In general, in d dimensions, the density of states is given by
g(E) =
V vol(S
d−1
)
2 · π
d/2
m
2π~
2
d/2
E
d/2−1
.
All these derivations work if we have a non-relativistic free particle, since we
assumed the dispersion relation between E and k, namely
E =
~
2
|k|
2
2m
.
For a relativistic particle, we instead have
E =
p
~
2
|k|
2
c
2
+ m
2
c
4
.
In this case,
|k|
is still quantized as before, and repeating the previous argument,
we find that
g(E) =
V E
2π
3
~
3
c
3
p
E
2
− m
2
c
4
.
We will be interested in the special case where m = 0. Then we simply have
g(E) =
V E
2
2π
3
~
3
c
3
.
We can start doing some physics with this.