Black-Body Radiation

Hamiltonian of Photons

The Hamiltonian for the EM field can be weitten as a sum of harmonic oscillators \begin{equation} H = \frac{1}{2}\sum_{\vec k, \alpha}[|p_{\vec k,\alpha}|^2 + \omega_\alpha^2(\vec k) + |u_{\alpha}(\vec k)|] \end{equation} With periodic boundary conditions, the allowed values of EM energy must be quantized according to the Hamiltonian \begin{equation} H^q = \sum_{\vec k,\alpha} \hbar c k(n_\alpha(\vec k) +\frac{1}{2}), n_\alpha(\vec k) = 0,1,2,\cdot\cdot\cdot \end{equation}

Internal Energy

The excitation energy (without zero-point energy) is \begin{equation} \frac{U}{V}=\frac{\hbar c}{\pi^2}(\frac{k_B T}{\hbar c})^4 \int_{0}^{\infty}\frac{x^3 dx}{e^x -1} = \frac{\pi^2}{15}(\frac{k_B T}{\hbar c})^3 k_B T \end{equation}

Partition Function

The partition function is \begin{equation} \Xi = \sum_{n_\alpha(\vec k)} \prod_{\vec k, \alpha} \text{exp} [-\beta\hbar\omega(\vec k)(n_\alpha(\vec k)+\frac{1}{2})] = \prod_{\vec k,\alpha} \frac{e^{-\beta\hbar ck/2}}{1-e^{-\beta\hbar ck}} \end{equation}

Other Physical Quantities

The free energy is \begin{equation} F=-k_B t\ln \Xi = 2V\int\frac{d^3 \vec k}{(2\pi)^3}[\frac{\hbar ck}{2}+k_B T\ln (1-e^{-\beta\hbar ck})] \end{equation} The pressure of photon gas is \begin{equation} P=-(\frac{\partial F}{\partial V})_T=P_0 + \frac{1}{3}\frac{E}{V} \end{equation}

Energy Flux Density

The escaping energy flux per unit area and per unit time from a hole is \begin{equation} \phi = \langle c_\bot \rangle\frac{E}{V} = \frac{c}{4}\frac{E}{V}=\frac{\pi^2}{60}\frac{k_B^4 T^4}{\hbar^3 c^3} \end{equation} The rsult, \(\phi = \sigma T\), is the Stefan’s law and \begin{equation} \sigma = \frac{\pi^2}{60}\frac{k_B^4}{\hbar^3 c^3} \approx 5.67\times10^{-8}W m^{-2}K^{-4} \end{equation} is the Stefan’s constant.

Planck Distribution

The energy density per unit frequency interval is \begin{equation} u(\omega,T) = \frac{\hbar}{\pi^2 c^3}\frac{\omega^3}{e^{\beta \hbar \omega}-1} \end{equation} This is the Planck distribution.