Blackbody Radiation

Every object above absolute zero radiates electromagnetic energy. A blackbody is the idealized limit: an object that absorbs all radiation that falls on it and emits a spectrum that depends only on its temperature, not on what it is made of. Furnace cavities, the Sun's photosphere, and the cosmic microwave background are all well approximated as blackbodies.

What experiment shows

When you measure the power radiated at each frequency $\nu$ by a blackbody at temperature $T$ , you find a smooth curve — the Planck spectrum — with two clear features:

The total power grows steeply with temperature; Stefan and Boltzmann measured this empirically and the result $P \propto T^4$ is called the Stefan–Boltzmann law.
The peak of the spectrum shifts to higher frequencies as temperature rises; Wien's displacement law states that the peak wavelength $\lambda_{\max}$ satisfies $\lambda_{\max} T = b$ , where $b \approx 2.898 \times 10^{-3}\,\text{m·K}$ .

Both of these are purely observational facts that 19th-century physics could fit with empirical formulas. The deeper question was: why does the spectrum have this shape?

The classical failure

Classical physics models the electromagnetic field inside a cavity as a collection of standing waves, each acting like a harmonic oscillator. Statistical mechanics says each such oscillator carries an average energy $k_B T$ at temperature $T$ (the equipartition theorem). The number of standing-wave modes with frequencies between $\nu$ and $\nu + d\nu$ grows as $\nu^2$ . Multiplying mode count by energy per mode gives the Rayleigh–Jeans law:

$u(\nu, T) = \frac{8\pi \nu^2}{c^3} k_B T.$

This formula agrees with experiment at low frequencies, but it predicts that $u(\nu, T)$ grows without limit as $\nu \to \infty$ . The total energy integrated over all frequencies diverges. This nonsensical prediction was named the ultraviolet catastrophe by Paul Ehrenfest in 1911 — the point where classical physics breaks down completely.

Setting up the quantum solution

In 1900, Max Planck found a formula that fits the observed spectrum exactly:

$u(\nu, T) = \frac{8\pi h \nu^3}{c^3} \cdot \frac{1}{e^{h\nu / k_B T} - 1}.$

Here $h \approx 6.626 \times 10^{-34}\,\text{J·s}$ is Planck's constant, a new fundamental constant of nature. At low frequencies ( $h\nu \ll k_B T$ ), the exponential can be expanded: $e^x - 1 \approx x$ for small $x$ , so the denominator becomes $h\nu / k_B T$ and the formula reduces to the Rayleigh–Jeans result. At high frequencies ( $h\nu \gg k_B T$ ), the exponential in the denominator grows rapidly, cutting off the spectrum and removing the catastrophe.

To derive this formula rather than just guess it, Planck had to assume that each oscillator in the cavity wall can only exchange energy with the field in discrete packets of size $\epsilon = h\nu$ . An oscillator at frequency $\nu$ can hold energy $0, h\nu, 2h\nu, 3h\nu, \ldots$ — no in-between values are allowed. This was the first statement that energy is quantized, and it launched the entire quantum revolution.

The reason this assumption kills the ultraviolet catastrophe is elegant: high-frequency modes require large energy quanta to be excited at all. At a given temperature $T$ , the thermal energy scale is $k_B T$ . When $h\nu \gg k_B T$ , the probability of exciting even the first level of a high-frequency mode is exponentially small, so those modes contribute almost nothing to the spectrum. Equipartition fails because energy cannot flow continuously into high-frequency modes — it must arrive in chunks too large for thermal fluctuations to supply.

This lesson sets the stage: the next lessons will show how Planck's quantum hypothesis leads directly to Einstein's photon picture, the photoelectric effect, and the full framework of quantum mechanics.