Prequisites: The Partition Function, The Boltzmann Distribution
Planck’s radiation law describes the spectral radiance of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature \(T\). It gives the power emitted per unit area, per unit solid angle, per unit frequency (or wavelength).
The Formula
In terms of frequency \(\nu\):
\[ B_\nu(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu / k_B T} - 1} \]
In terms of wavelength \(\lambda\):
\[ B_\lambda(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc / \lambda k_B T} - 1} \]
where: - \(h\) is Planck’s constant (\(6.626 \times 10^{-34}\) J·s) - \(c\) is the speed of light - \(k_B\) is Boltzmann’s constant - \(T\) is the absolute temperature
Physical Content
The law resolves the ultraviolet catastrophe that plagued classical physics. The Rayleigh-Jeans law, derived from classical equipartition, predicts \(B_\nu \propto \nu^2 T\), which diverges as \(\nu \to \infty\). Planck’s key insight was that energy exchange between matter and radiation occurs in discrete quanta of size \(E = h\nu\).
The Bose-Einstein factor \((e^{h\nu/k_BT} - 1)^{-1}\) gives the mean occupation number of photon modes at frequency \(\nu\). At low frequencies (\(h\nu \ll k_B T\)), this approaches \(k_B T / h\nu\), recovering the classical Rayleigh-Jeans regime. At high frequencies (\(h\nu \gg k_B T\)), the exponential suppression cuts off the spectrum, preventing the divergence.
Derivation Sketch
Consider a cavity with standing electromagnetic modes. Each mode of frequency \(\nu\) is a quantum harmonic oscillator with energy levels \(E_n = nh\nu\) (taking the zero-point energy as the reference). The partition function is:
\[ Z = \sum_{n=0}^{\infty} e^{-nh\nu / k_B T} = \frac{1}{1 - e^{-h\nu / k_B T}} \]
The mean energy per mode is:
\[ \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} = \frac{h\nu}{e^{h\nu / k_B T} - 1} \]
Multiplying by the density of states \(g(\nu) = 8\pi\nu^2/c^3\) and converting to spectral radiance yields Planck’s formula.
Limiting Cases and Consequences
Wien’s displacement law follows from finding the maximum of \(B_\lambda\):
\[ \lambda_{\text{max}} T = \frac{hc}{k_B} \cdot \frac{1}{W(5e^5/e)} \approx 2.898 \times 10^{-3} \text{ m·K} \]
where \(W\) is the Lambert \(W\) function (from solving \(5 = x/(1 - e^{-x})\)).
Stefan-Boltzmann law follows from integrating over all frequencies:
\[ j = \int_0^\infty B_\nu \, d\nu = \sigma T^4, \quad \sigma = \frac{2\pi^5 k_B^4}{15 h^3 c^2} \]