The Photon Gas and Planck's Radiation Law

Thermal radiation in a cavity held at temperature $T$ is a gas of photons. The walls emit and absorb photons continuously, so their number is not fixed by any conservation law but set by the condition of thermal equilibrium. The general framework for ideal quantum gases converts a sum over occupied single-particle modes into an integral weighted by the Bose-Einstein occupation; for photons that framework specializes cleanly because two features fix the inputs at once — the dispersion is linear, $\epsilon =\hslash \omega =c\hslash k$, and the chemical potential is zero. The output is the Planck law, the first quantitative success of the quantum hypothesis and the spectrum of every blackbody from a furnace to the cosmic microwave background.

Non-conservation and the vanishing chemical potential

Photons are spin-$1$ massless bosons. A cavity wall at temperature $T$ absorbs a photon into an excited atomic state and re-emits one, and nothing constrains the total count $N$: the equilibrium value of $N$ is whatever minimizes the free energy. For a system in contact with a heat bath at fixed $T$ and $V$, the equilibrium state minimizes the Helmholtz free energy $F(T,V,N)$ over every variable free to adjust. If $N$ is free, the minimum condition is

The left-hand side is the definition of the chemical potential, $\mu =(\partial F/\partial N{)}_{T,V}$, so equilibrium of a non-conserved species forces

Every occupation number then reduces to the Bose-Einstein form evaluated at $\mu =0$,

with no fugacity to determine.¹ The same argument applies to any gas of non-conserved quasiparticles, and the next lesson applies it verbatim to phonons.

Counting the cavity modes

The allowed field configurations in a cubical cavity of side $L$ are standing waves that vanish on the walls. Each Cartesian component of the wavevector is quantized,

so the allowed modes are the points of a cubic lattice in the positive octant of $\vec{k}$-space, spaced $\pi /L$ apart. A single standing wave carries a definite number of half-wavelengths along each axis.

The number of modes with wavevector magnitude up to $k$ is the volume of the octant of a sphere of radius $n=kL/\pi$ in the integer lattice, one lattice point per unit cell, multiplied by two for the polarizations:

Differentiating gives the mode density in $k$, and converting through $\omega =ck$ and $\omega =2\pi f$ gives the density in frequency,

The factor $f^2$ is the geometric growth of the number of modes with frequency, the same factor that drives the classical divergence below.

The Planck spectral density

Each mode of frequency $f$ is an independent harmonic oscillator whose quantized excitations are photons. The mean energy stored in a mode, measured from its ground state, is the occupation times the quantum $hf$,

The spectral energy density — energy per unit volume per unit frequency — is the mode density per volume times the mean energy per mode:

The two forms describe the same radiation field but peak at different arguments, because the Jacobian $\mathbf{d}f/\mathbf{d}\lambda =-c/{\lambda }^2$ reshapes the curve.²

Classical limits and the ultraviolet catastrophe

The mean energy per mode carries the whole quantum content. Expanding it in the two limits recovers the classical law and its failure.

• Rayleigh-Jeans limit ($hf\ll k_{B}T$). The denominator linearizes, $e^{hf/k_{B}T}-1\approx hf/k_{B}T$, so $\bar{\epsilon }\to k_{B}T$ and $$u(f)\to \frac{8\pi f^2}{c^3}{k}_{B}T.$$ This is exactly equipartition — $k_{B}T$ per electromagnetic mode — and it is the classical prediction. It grows without bound as $f^2$, so the total energy density ${\int }_0^{\infty }u(f)\mathbf{d}f$ diverges. That divergence is the ultraviolet catastrophe: classical physics assigns $k_{B}T$ to arbitrarily high-frequency modes, of which there are infinitely many.
• Wien limit ($hf\gg k_{B}T$). The $-1$ is negligible against the exponential, so $$u(f)\to \frac{8\pi h{f}^3}{c^3}{e}^{-hf/k_{B}T}.$$ The exponential suppression reflects the cost of exciting even one photon in a mode whose quantum $hf$ exceeds the thermal energy: such modes are frozen out, their mean occupation exponentially small.

The quantization $\bar{\epsilon }=hf/(e^{hf/k_{B}T}-1)$ interpolates between the two. At low frequency it returns $k_{B}T$; past $hf\approx k_{B}T$ it collapses toward zero and tames the $f^2$ growth of the mode count. The finite peak is the balance of a rising mode density against a falling occupation.

Wien's displacement law

The spectrum peaks at a frequency proportional to temperature. Writing $x=hf/k_{B}T$, the frequency density is $u\propto x^3/(e^x-1)$, and setting its derivative to zero gives the transcendental condition

with root $x_{\max }\approx 2.821$. The peak frequency is therefore

Repeating the calculation for the wavelength density $u(\lambda )\propto {\lambda }^{-5}/(e^{hc/\lambda k_{B}T}-1)$ gives the condition $5(1-e^{-y})=y$ with $y=hc/\lambda k_{B}T$ and root $y_{\max }\approx 4.965$, so

The two peaks disagree, $x_{\max }\ne y_{\max }$, precisely because of the Jacobian between $f$ and $\lambda$; the peak of the frequency curve and the peak of the wavelength curve are different physical features of the same field.³

Summary

• Photon number is not conserved; equilibrium at fixed $T,V$ minimizes $F$ over $N$, giving $\mu =(\partial F/\partial N{)}_{T,V}=0$ and Bose occupation $⟨n⟩=1/(e^{hf/k_{B}T}-1)$.
• Standing-wave modes in a cavity number $\mathbf{d}\mathcal{N}/\mathbf{d}f=8\pi V{f}^2/c^3$ after counting two polarizations; the $f^2$ growth is the geometric origin of the classical divergence.
• The Planck density $u(f)=(8\pi h{f}^3/c^3)/(e^{hf/k_{B}T}-1)$ returns the Rayleigh-Jeans law $8\pi f^2{k}_{B}T/c^3$ for $hf\ll k_{B}T$ and the Wien tail for $hf\gg k_{B}T$; the Bose factor cuts off the ultraviolet catastrophe.
• The peak obeys Wien's displacement law: $f_{\max }\propto T$, and ${\lambda }_{\max }T=2.898\mathrm{mm}\mathrm{K}$, the frequency and wavelength peaks differing by the Jacobian between them.