Advanced Solid State Physics

To get some macroscopic properties the partition function is useful, especially the canonical partition
functionZ(T,V)

Z(T,V) =

∑

q

e−

Eq

kBT,

which is relevant at a constant temperatureT. For photons the sum over all microstates is:

Z(T,V) =

∑

j 1

...

∑

jsmax

e

−kB^1 T

∑smax

s=1 ~ωs(js+

(^12) )
The sum in the exponent can be written as a product:

Z(T,V) =

∑

j 1

...

∑

jsmax

s∏max

s=

e−

~ωs(js+^12 )

kBT

This is a sum ofjsmaxmaxterms, each term consisting ofsmaxexponential factors. It isn’t obvisious, but
this sum can also be written as:

Z(T,V) =

s∏max

s=

[

e

− 2 ~kωBsT

+e

− 23 k~BωsT

+e

− 25 k~BωsT

+...

]

Now if the right factor is put out, the form of a geometric series (1 +^1 x+x^12 +...= 1 −^1 x) is derived:

Z(T,V) =

s∏max

s=



 e

~ωs

2 kBT

1 −e

~ωs

kBT





The free energyFis given by

F=−kBTln(Z).

InsertingZleads to

F=

∑

s

~ωs

2

+kBT

∑

s

ln

(

1 −e−

~ωs

kBT

)

. (33)

The first term is the ground state energy, which cannot vanish. Often it is neglected, like in the
following calculations.