---
title: Blackbody Radiation and the Planck Quantum
module: Origins of the Quantum
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  Millikan's oil-drop experiment fixed the electron charge as an indivisible
  unit, and the spectrum of thermal radiation forced a second, deeper quantum.
  Classical physics predicts an infinite energy density at short wavelengths;
  Planck removed the divergence by allowing a cavity oscillator to hold only
  energies that are integer multiples of hf, the first appearance of the quantum
  of action.
topics: [Origins of the Quantum]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 3 — Quantization of Charge, Light, and Energy; §3-1 Quantization of Electric Charge, §3-2 Blackbody Radiation"
  - book: Tipler & Mosca
    ref: "Ch. 34 §34-1 Wave-Particle Duality"
draft: false
---

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,

$$
F = N_A e,
$$

with $F \approx 96{,}500\ \text{C}$ measurable but $N_A$ and $e$ separately
unknown in Faraday's time.[^tl-charge] 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

$$
\frac{q}{m} = \frac{u}{RB},
$$

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.[^tl-millikan]

$$
% caption: Millikan's oil-drop apparatus. A charged drop between capacitor
% plates is watched through a telescope; its terminal fall and field-driven
% rise fix the charge it carries.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % atomizer
  \draw[black] (-3.6,2.3) -- (-2.9,2.3) -- (-3.0,2.05) -- (-3.5,2.05) -- cycle;
  \node[anchor=south, font=\scriptsize] at (-3.25,2.35) {atomizer};
  \draw[->, black] (-2.9,2.18) -- (-2.1,2.0);
  % top plate with hole
  \draw[acc, very thick] (-2.0,1.6) -- (-0.15,1.6);
  \draw[acc, very thick] (0.15,1.6) -- (2.0,1.6);
  \node[anchor=south east, font=\scriptsize, text=acc] at (2.0,1.6) {$+$};
  % bottom plate
  \draw[acc, very thick] (-2.0,-1.2) -- (2.0,-1.2);
  \node[anchor=north east, font=\scriptsize, text=acc] at (2.0,-1.2) {lower plate};
  % battery symbol
  \draw[black] (2.0,1.6) -- (2.9,1.6) -- (2.9,-1.2) -- (2.0,-1.2);
  \draw[black] (2.75,0.55) -- (3.05,0.55);
  \draw[black, very thick] (2.82,0.25) -- (2.98,0.25);
  \draw[black] (2.75,-0.05) -- (3.05,-0.05);
  \node[anchor=west, font=\scriptsize] at (3.1,0.2) {$\mathcal{E}$};
  % drops
  \fill[acc] (0.0,1.15) circle (1.6pt);
  \fill[acc] (-0.6,0.4) circle (1.6pt);
  \fill[acc] (0.5,-0.1) circle (1.6pt);
  \fill[acc] (-0.2,-0.6) circle (1.6pt);
  % forces on one drop
  \draw[->, black, thick] (0.5,-0.1) -- (0.5,0.7);
  \node[anchor=west, font=\scriptsize] at (0.55,0.55) {$q\mathcal{E}$};
  \draw[->, black, thick] (0.5,-0.1) -- (0.5,-0.9);
  \node[anchor=west, font=\scriptsize] at (0.55,-0.8) {$mg$};
  % telescope
  \draw[black] (-4.2,-0.3) -- (-3.2,0.0) -- (-3.2,-0.6) -- cycle;
  \draw[black, dashed] (-3.2,-0.3) -- (-0.6,0.4);
  \node[anchor=east, font=\scriptsize] at (-4.2,-0.45) {telescope};
  % light source
  \draw[black] (3.9,-0.6) circle (0.18);
  \node[anchor=west, font=\scriptsize] at (4.1,-0.6) {light};
  \draw[->, black] (3.75,-0.55) -- (0.7,-0.1);
\end{tikzpicture}
$$

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:

> **Definition (Elementary charge).** The charge on any isolated body is an
> integer multiple $q = ne$ of the **elementary charge**
> $e = 1.602176 \times 10^{-19}\ \text{C}$. No smaller free charge has been
> observed. Charge, like matter, is granular rather than continuous.

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.

> **Definition (Blackbody).** An ideal **blackbody** absorbs all radiation
> incident on it. Its emitted spectrum depends only on its absolute temperature
> $T$, not on the material, shape, or surface color. A small hole into a cavity
> is the best laboratory realization: radiation entering it is absorbed by the
> walls before it can find its way back out.

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:

$$
R = \sigma T^4,
\qquad
\sigma = 5.6703 \times 10^{-8}\ \text{W/m}^2\text{K}^4.
$$

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:

$$
\lambda_m T = 2.898 \times 10^{-3}\ \text{m}\cdot\text{K}.
$$

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 $R(\lambda)$.

$$
% caption: Measured spectral radiance of a blackbody at several temperatures.
% The area under each curve grows as the fourth power of T; the peak shifts to
% shorter wavelength as T rises (Wien's law).
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,0) -- (7.6,0) node[anchor=north east] {wavelength};
  \draw[->, black] (0,0) -- (0,4.4) node[anchor=south east, align=left] {spectral\\radiance};
  % low T curve (broad, low, peak far right)
  \draw[acc, thick, smooth] plot coordinates
    {(0.2,0.02)(0.9,0.15)(1.8,0.55)(2.8,0.85)(3.6,0.95)(4.5,0.82)(5.6,0.55)(6.8,0.32)(7.4,0.24)};
  \node[acc, anchor=west, font=\scriptsize] at (5.8,0.72) {$T_1$};
  % mid T
  \draw[black, thick, densely dashed, smooth] plot coordinates
    {(0.15,0.05)(0.7,0.5)(1.4,1.4)(2.1,1.95)(2.7,2.05)(3.5,1.7)(4.6,1.05)(5.8,0.6)(7.0,0.35)};
  \node[black, anchor=west, font=\scriptsize] at (3.6,1.75) {$T_2$};
  % high T (tall, narrow, peak left)
  \draw[black!70, thick, densely dotted, smooth] plot coordinates
    {(0.1,0.1)(0.55,1.2)(1.05,2.9)(1.5,3.9)(1.95,3.7)(2.6,2.6)(3.5,1.4)(4.8,0.7)(6.4,0.35)};
  \node[black!70, anchor=west, font=\scriptsize] at (1.9,3.85) {$T_3$};
  % peak markers descending
  \draw[black, dashed] (1.5,3.9) -- (1.5,0) node[anchor=north, font=\scriptsize] {};
  \draw[black, dashed] (2.7,2.05) -- (2.7,0);
  \draw[black, dashed] (3.6,0.95) -- (3.6,0);
  \node[font=\scriptsize, anchor=south] at (2.55,4.05) {$T_3 > T_2 > T_1$};
\end{tikzpicture}
$$

**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 $L = 4\pi r^2 \sigma T^4$,

$$
r^2 = 100\, r_\odot^2 \left(\frac{T_\odot}{T_\text{star}}\right)^4
\;\Rightarrow\;
r = 10\, r_\odot \left(\frac{5800}{3000}\right)^2 = 37.4\, r_\odot.
$$

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 $R = \tfrac{1}{4}cU$, and the same factor relates the two
spectral distributions,

$$
R(\lambda) = \tfrac{1}{4}\, c\, u(\lambda).
$$

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 $\d\lambda$, and how much energy each mode carries on average.

$$
% caption: A cavity with a pinhole is an ideal blackbody. Standing
% electromagnetic waves fit an integer number of half-wavelengths between the
% walls; shorter wavelengths pack in more modes.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % cavity box
  \draw[black, thick] (0,0) rectangle (5.0,3.0);
  \fill[white] (0,1.35) rectangle (0.08,1.65);
  \draw[acc, very thick] (-0.05,1.35) -- (-0.05,1.65);
  \node[anchor=east, font=\scriptsize, text=acc] at (-0.15,1.5) {hole};
  % incoming ray absorbed by bouncing
  \draw[->, black] (-1.0,1.9) -- (0.0,1.6);
  \draw[black] (0.0,1.6) -- (2.2,2.9);
  \draw[black] (2.2,2.9) -- (3.6,0.1);
  \draw[black] (3.6,0.1) -- (4.9,1.7);
  \draw[black] (4.9,1.7) -- (3.2,2.9);
  % standing wave modes on the right
  \draw[acc, thick, smooth] plot coordinates
    {(0.4,0.55)(1.15,0.85)(1.9,0.55)} ;
  \node[font=\scriptsize, anchor=north] at (1.15,0.5) {one mode};
  \draw[acc, thick, smooth] plot coordinates
    {(2.7,0.55)(2.95,0.8)(3.2,0.55)(3.45,0.3)(3.7,0.55)(3.95,0.8)(4.2,0.55)};
  \node[font=\scriptsize, anchor=north] at (3.45,0.28) {shorter, more modes};
\end{tikzpicture}
$$

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

$$
n(\lambda)\, \d\lambda = \frac{8\pi}{\lambda^4}\, \d\lambda.
$$

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,

$$
u(\lambda) = kT\, n(\lambda) = \frac{8\pi kT}{\lambda^4}.
$$

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,

$$
U = \int_0^\infty u(\lambda)\, \d\lambda \to \infty.
$$

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.

$$
% caption: The Rayleigh-Jeans law tracks the data at long wavelengths but climbs
% without bound toward short wavelengths; Planck's law bends over and matches the
% measured curve everywhere.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[anchor=north east] {wavelength};
  \draw[->, black] (0,0) -- (0,4.5) node[anchor=south east, align=left] {energy\\density};
  % Rayleigh-Jeans: diverges as lambda -> 0
  \draw[black, thick, densely dashed, smooth] plot coordinates
    {(6.6,0.32)(5.2,0.5)(4.0,0.85)(3.0,1.45)(2.2,2.4)(1.6,3.5)(1.25,4.35)};
  \node[black, anchor=west, font=\scriptsize] at (1.25,4.2) {Rayleigh-Jeans};
  % Planck: bends over, matches data
  \draw[acc, thick, smooth] plot coordinates
    {(0.2,0.1)(0.7,0.9)(1.2,2.2)(1.7,3.0)(2.2,3.05)(2.9,2.55)(3.8,1.75)(5.0,0.95)(6.6,0.42)};
  \node[acc, anchor=west, font=\scriptsize] at (2.9,2.75) {Planck};
  % data points on the Planck curve
  \foreach \p in {(0.7,0.9),(1.7,3.0),(2.9,2.55),(3.8,1.75),(5.0,0.95),(6.6,0.42)}
    \fill[black] \p circle (1.6pt);
  \node[font=\scriptsize, anchor=west] at (5.1,1.25) {data};
\end{tikzpicture}
$$

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.

> **Postulate (Energy quantization).** A cavity oscillator of frequency $f$ can
> hold energy only in integer multiples of a fixed quantum,
> $$
> E_n = n\varepsilon = nhf, \qquad n = 0, 1, 2, \dots
> $$
> where $h$ is a new constant of nature. Energy is exchanged with the radiation
> field in whole quanta, never in fractions.

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:

$$
\sum_{n=0}^\infty x^n = \frac{1}{1-x},
\qquad
\sum_{n=0}^\infty n\, x^n = \frac{x}{(1-x)^2}.
$$

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

$$
\bar{E} = \varepsilon\, \frac{\sum_n n x^n}{\sum_n x^n}
= \varepsilon\, \frac{x}{1-x}
= \frac{hf}{e^{hf/kT} - 1}.
$$

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

$$
% caption: Average energy per mode. Classical equipartition holds it flat at kT
% for every frequency; Planck's value falls off once the quantum hf exceeds the
% thermal energy kT, starving the short-wavelength modes.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[anchor=north east, align=center] {quantum size $hf$};
  \draw[->, black] (0,0) -- (0,3.6) node[anchor=south east, align=left] {average\\energy};
  % classical kT flat line
  \draw[black, thick, densely dashed] (0,2.6) -- (7.0,2.6);
  \node[black, anchor=south west, font=\scriptsize] at (4.1,2.6) {classical: $kT$};
  \draw[black, dashed] (0,2.6) -- (0.0,2.6);
  \node[anchor=east, font=\scriptsize] at (-0.05,2.6) {$kT$};
  % Planck curve falling from kT
  \draw[acc, thick, smooth] plot coordinates
    {(0.15,2.5)(0.7,2.15)(1.4,1.65)(2.2,1.15)(3.1,0.75)(4.2,0.42)(5.4,0.22)(6.8,0.1)};
  \node[acc, anchor=west, font=\scriptsize] at (2.3,1.35) {Planck};
  % marker where hf = kT
  \draw[black, dashed] (2.55,0) -- (2.55,1.0);
  \node[anchor=north, font=\scriptsize] at (2.55,0) {$hf = kT$};
\end{tikzpicture}
$$

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**:

> **Theorem (Planck's law).** The spectral energy density of blackbody radiation
> is
> $$
> u(\lambda) = \frac{8\pi h c}{\lambda^5}\,
> \frac{1}{e^{hc/\lambda kT} - 1}.
> $$
> It agrees with the measured spectrum at every temperature, and the constant
> $h = 6.626 \times 10^{-34}\ \text{J}\cdot\text{s} = 4.136 \times 10^{-15}\
> \text{eV}\cdot\text{s}$ is fixed by the fit.

### 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 $hc/\lambda kT$.

| Regime | Condition | Denominator | $u(\lambda)$ | Recovers |
| --- | --- | --- | --- | --- |
| Long wavelength | $hc \ll \lambda kT$ | $e^{hc/\lambda kT} - 1 \approx \dfrac{hc}{\lambda kT}$ | $\dfrac{8\pi kT}{\lambda^4}$ | Rayleigh-Jeans |
| Short wavelength | $hc \gg \lambda kT$ | $e^{hc/\lambda kT} - 1 \approx e^{hc/\lambda kT}$ | $8\pi hc\, \lambda^{-5} e^{-hc/\lambda kT}$ | Wien's exponential cutoff |

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 $\d u/\d\lambda = 0$ 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

$$
\lambda_m = \frac{2.898 \times 10^{-3}\ \text{m}\cdot\text{K}}{5800\ \text{K}}
= 500\ \text{nm},
$$

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

$$
\bar{E} = \frac{hf}{e^{hf/kT} - 1} = \frac{kT}{e - 1} = 0.582\, kT,
$$

already noticeably below the classical $kT$. For $hf = 10\, kT$ 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.

$$
% caption: Two constants of nature, two granularities. Millikan fixed the lump of
% charge, e = 1.602e-19 C; Planck fixed the lump of action, h = 6.626e-34 J s,
% tying a quantum's energy to its frequency.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[draw, thick, minimum width=34mm, minimum height=15mm, align=center]
    (q1) at (0,0) {charge is granular\\$q = ne$};
  \node[draw, thick, draw=acc, text=acc, minimum width=34mm, minimum height=15mm, align=center]
    (q2) at (5.2,0) {action is granular\\$E = nhf$};
  \node[align=center, font=\scriptsize] (m1) at (0,-2.0) {Millikan\\elementary charge $e$};
  \node[align=center, font=\scriptsize, text=acc] (m2) at (5.2,-2.0) {Planck\\quantum of action $h$};
  \draw[->, black] (q1) -- (m1);
  \draw[->, acc] (q2) -- (m2);
\end{tikzpicture}
$$

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](/quantum-mechanics/old-quantum-theory/the-photoelectric-effect-and-the-photon),
the subject of the next lesson. The same constant $h$ then reappears in the
[Compton scattering](/quantum-mechanics/old-quantum-theory/x-rays-and-the-compton-effect)
of X-rays, in
[Bohr's model of hydrogen](/quantum-mechanics/old-quantum-theory/the-old-quantum-theory-bohr-and-sommerfeld), and
throughout the quantum theory that follows.

[^tl-charge]: Tipler & Llewellyn, §3-1 — Faraday's electrolysis law $F = N_A e$ and Thomson's crossed-field measurement of $e/m$ for cathode rays.
[^tl-millikan]: Tipler & Llewellyn, §3-1 — Millikan's oil-drop experiment; charges observed always as integer multiples of $e = 1.602176 \times 10^{-19}\ \text{C}$.