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.