So it is∝T^4 .σ=^2 π
(^5) k (^4) B
15 h^3 c^2 = 5.^67 ·^10
− (^8) Wm− (^2) K− (^4) is the Stefan-Boltzmann constant. TheentropyS
is
S=−
∂F
∂T
=
16 σV T^3
3 cJ/K.
This is the fermi expression for the entropy for a gas of photons at a specific temperature. By knowing
the free energy F and the entropy S theinternal energyU can be calculated:
U=F+TS=
4 σV T^4
cJ.
From these expressions one can calulate some more measurable quantities. An important quantity is
thepressureP.
P=−∂F
∂V
=
4 σT^4
3 cN/m^2.A photon at a specific temperature exerts a pressure. Furthermore photons carrymomentump,
even though they have no mass.
p=~kThe momentum is related to the wavelength. Thespecific heatCV is also easy to calculate:
CV=−∂U
∂T
=
16 σV T^3
cJ/K (34)
It is the amount of energy in Joule you need to raise the temperature of some volume of material by
1 Kelvin. The specific heat is∝T^3. Eqn. 34 is also the Debye form for phonons. So phonons also
have energy. (If you heat up any kind of solid, some energy goes in the lattice vibration)
The difference between phonons and photons is that for photons it is∝T^3 for arbitrary high tem-
peratures (its just always∝T^3 ) and for phonons there is a debye frequency (a highest frequency of
soundwaves in a solid), which has to do with the spacing between the atoms. So in terms ofsound
waveseqn. 34 is only thelow temperature form, but forphotonsit is forall temperatures.
4.1.1 Recipe for the Quantization of Fields
- The first thing to do is to find the classical normal modes. If it is some kind of linear wave
equation plane waves will always work (so try plane waves). If the equations are nonlinear,
linearize the equations around the point you are interested in. - Quantize the normal modes and look what the energies are.
- Calculate the density of modes in terms of frequency. (D(k) is always the same)
- Quantize the modes
- Knowing the distribution of the quantum states, deduce thermodynamic quantities.
Z(T,V) =∑
qexp(
−
Eq
kBT)F=−kBTlnZ