Photon Gas

Topics

• Planck Distribution Function
• Blackbody Radiation
• Free Electron Gas

$⟨\epsilon ⟩=⟨s⟩\hslash \omega$

• $\epsilon$ is the average energy for the frequency mode $\omega$
• $⟨s⟩$ can be found by the Planck distribution function

$⟨\epsilon ⟩=\hslash {\omega }_0\left(⟨s⟩+\frac{1}{2}\right)=\frac{1}{2}\coth \left(\frac{\hslash \omega }{\tau }\right)$

• Expanded version of above…

$H=pc=\hslash kc=\hslash c\left(\frac{\pi }{L}\sqrt{n_x^2+n_y^2+n_z^2}\right)$

• The energy of a photon gas in a cavity

$U=-\sum _n{s}_n\hslash {\omega }_n$

• Internal energy of a photon gas
• $s_n$ is the number of photons in mode ${\omega }_n$
• ${\omega }_n$ is the angular frequency for energy mode $n$

$Z=\frac{1}{1-e^{{}_{\hslash \omega \beta }}}$

• [Partition function] for photon gas