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\sect{Harmonic oscillators, Photons}

Find the state equations of photon gas in 1D/2D/3D cavity
within the framework of the canonical formalism, 
regarding the electromagnetic modes as a collection 
of harmonic oscillators.  The volume of the cavity 
is $L^d$ with ${d=1,2,3}$. The temperature is $T$. 

\Dn

\begin{itemize}
\item[(1)] Write the partition function for a single mode $\omega$.
\item[(2)] Find the mode average occupation $f(\omega)$.
\item[(3)] Find the spectral density of modes $g(\omega)$. 
\item[(4)] Find the energy $E(T)$ of the photon gas.
\item[(5)] Find the free energy $F(T)$ of the photon gas.
\item[(6)] Find an expression for the pressure $P(T)$ of the photon gas.
\end{itemize}
  
\Dn

Note: additional exercises on photon gas and blackbody radiation 
can be found in the context of quantum gases.  
Formally, photon gas is like Bose gas with chemical potential ${\mu=0}$.
Note that the same type of calculation appears 
in Debye model ("acoustic" phonons instead of "transverse" photons). 
