Blackbody Radiation and the Planck Quantum

Classical physics is continuous. Energy, charge, and the intensity of a light beam were all quantities you could subdivide without limit. Between 1897 and 1905 three measurements broke that assumption, one after another: electric charge comes in a smallest lump, the energy exchanged between light and matter comes in lumps, and the energy of a mechanical oscillator comes in lumps. This lesson takes the first and third. Charge quantization was expected — matter was already known to be atomic — and it fell to a direct measurement. Energy quantization was not expected, and it emerged from a problem no one could solve with continuous physics: the spectrum of thermal radiation.

The elementary charge

Faraday's electrolysis law relates the charge $F$ that deposits one gram-ionic weight of a monovalent ion to Avogadro's number,

with $F\approx 96,500\text{C}$ measurable but $N_A$ and $e$ separately unknown in Faraday's time.¹ The ratio $e/m$ of the electron came first, from J. J. Thomson's 1897 cathode-ray tube: crossed electric and magnetic fields set to leave the beam undeflected give the speed, $u=\mathcal{E}/B$, after which the magnetic deflection alone yields

where $R$ is the radius of the circular path in the field $B$. Thomson found $e/m\approx 1.76\times 10^{11}\text{C/kg}$, some 2000 times the value for the hydrogen ion, and the same for every gas and cathode metal — evidence of a single sub-atomic particle common to all matter.

The charge itself required a second experiment. Millikan sprayed fine oil drops into the air between the plates of a capacitor, charged them by friction in the spray nozzle (and by exposure to X-rays), and balanced or tracked a single drop against gravity with a vertical field.²

A drop falling at terminal velocity balances weight against Stokes drag, fixing its radius and mass. Turning on the field to hold it, or reversing the field to lift it, then gives the charge from the force balance $q\mathcal{E}=mg$ (in the suspended case). Every value Millikan measured was an integer multiple of one number:

Charge quantization was the expected discovery. The unexpected one was waiting in the light emitted by a warm body.

Thermal radiation

Every opaque body absorbs part of the radiation falling on it and re-emits radiation of its own. A good absorber is a good emitter: at thermal equilibrium the two rates match. The limiting case is a body that absorbs everything.

Two empirical laws describe the emission before any theory of its shape. The first, found by Stefan in 1879 and derived thermodynamically by Boltzmann, gives the total power radiated per unit area:

The fourth-power dependence is steep. Doubling the absolute temperature of a star raises its radiated power per unit area by $2^4=16$. The second law, found by Wien in 1893, locates the peak of the spectrum:

The wavelength of maximum emission shifts inversely with temperature — a cooling body reddens, a heating one whitens. Both laws appear directly in the measured family of spectral distribution curves $R(\lambda )$.

Example — the size of a star. A star with peak wavelength implying surface temperature $3000\text{K}$ radiates $100$ times the Sun's power. Wien's law gives both surface temperatures ($T_{\odot }=5800\text{K}$). Equating luminosities $L=4\pi r^2\sigma T^4$,

At $r_{\odot }=6.96\times 10^8\text{m}$ this star spans $2.6\times 10^{10}\text{m}$, roughly half Mercury's orbital radius — a red giant.

The cavity, its modes, and the classical prediction

The power radiated from the hole is proportional to the energy density $U$ inside the cavity, with $R=\frac{1}{4}cU$, and the same factor relates the two spectral distributions,

So the problem reduces to computing $u(\lambda )$, the electromagnetic energy per unit volume per unit wavelength standing inside the box. Classical physics splits that into two factors: how many modes of oscillation the cavity supports in the interval $\mathbf{d}\lambda$, and how much energy each mode carries on average.

The mode count per unit volume, independent of cavity shape, is

Classical equipartition assigns each mode the average energy of a one-dimensional oscillator, $\bar{E}=kT$, split equally between its kinetic and potential parts. Multiplying,

This is the Rayleigh-Jeans law. It matches experiment at long wavelengths and fails catastrophically at short ones: as $\lambda \to 0$ the mode count diverges, each mode still carries $kT$, and the total energy density is infinite,

Every warm object would hold infinite energy and radiate infinitely at short wavelengths. The name for the discrepancy was the ultraviolet catastrophe — catastrophe meant literally, since the divergence is in the integrated energy, not just the shape.

Both classical factors were correct on their own terms, and no one could tell in advance which one to blame. The mode count $n(\lambda )$ survived. Equipartition did not.

Planck's quantum hypothesis

Planck first found, by curve-fitting, a function $u(\lambda )$ that matched the data. He then worked backward to the assumption that would produce it. The change needed was in the average energy per mode: it must fall toward zero as $\lambda \to 0$ rather than stay pinned at $kT$. Planck obtained this by breaking with the continuum.

With energies discrete, the average over the Maxwell-Boltzmann population $f_n=A{e}^{-E_n/kT}$ becomes a sum rather than an integral. Writing $x=e^{-hf/kT}$, the normalization and the average are geometric series:

Their ratio, times the quantum $\epsilon =hf$, gives the average energy per mode:

This is the factor that classical physics got wrong. Compare the two side by side.

When the quantum is small compared with the thermal energy, $hf\ll kT$, the denominator expands as $e^{hf/kT}-1\approx hf/kT$ and $\bar{E}\to kT$ — the classical result is recovered for the long-wavelength modes. When the quantum is large, $hf\gg kT$, exciting even a single quantum is improbable, the exponential dominates, and $\bar{E}\to 0$. The short-wavelength modes are frozen out — the behavior the data demanded.

Multiplying the corrected average energy by the mode count, with $f=c/\lambda$, gives Planck's radiation law:

The classical limits recovered

Planck's law is not a separate formula bolted onto the classical one; it contains it. Take the two extremes of the dimensionless ratio $hc/\lambda kT$.

The short-wavelength form goes to zero, removing the ultraviolet catastrophe. The long-wavelength form is precisely Rayleigh-Jeans, which is why the classical law worked where it did. Integrating the full expression over all wavelengths reproduces the Stefan-Boltzmann $T^4$ law and fixes $\sigma$ in terms of $h$, $k$, and $c$; setting $\mathbf{d}u/\mathbf{d}\lambda =0$ reproduces Wien's displacement constant. Both empirical laws fall out of the single quantum hypothesis.

Example — the peak of the solar spectrum. For $T_{\odot }=5800\text{K}$, Wien's law gives

near the middle of the visible band — the Sun radiates most strongly in exactly the range our eyes evolved to use.

Example — a frozen mode. At the frequency where $hf=kT$, the average energy is

already noticeably below the classical $kT$. For $hf=10kT$ it is $\bar{E}\approx 5\times 10^{-4}kT$: the mode is effectively dead.

Interpretation of the quantum

Planck regarded the quantization as a property of the cavity oscillators — a mathematical device, not a statement about light itself — and spent years trying to reconcile it with classical physics. He did not succeed, because there was nothing to reconcile. The constant $h$ has units of energy times time, the units of action, and its appearance signals that action itself is granular at the atomic scale.

The person who took the quantum literally as a property of light was Einstein. Where Planck quantized the emitters, Einstein quantized the radiation field itself and used it to explain a phenomenon that had nothing to do with cavities — the photoelectric effect, the subject of the next lesson. The same constant $h$ then reappears in the Compton scattering of X-rays, in Bohr's model of hydrogen, and throughout the quantum theory that follows.