The ultraviolet catastrophe

What does classical physics predict for the spectrum of a black body? The calculation was carried out by Lord Rayleigh and James Jeans around 1900. The idea is to treat the radiation inside the cavity as a collection of standing waves (the “modes” of the electromagnetic field) and to assign to each mode, by the classical principle of equipartition of energy, the same average energy $k_B T$. Counting how many modes fall in each interval of wavelengths gives the spectral density

$u_{RJ}(\lambda,T) = \frac{8\pi\,k_B\,T}{\lambda^4}$

The formula works very well at long wavelengths (infrared, microwave), where it overlaps with the measured curve. But it hides a disaster: the factor $1/\lambda^4$ makes $u_{RJ}$ grow without limit as $\lambda \to 0$. The shorter the wavelength, the more energy classical theory assigns to the cavity, with no bound.

The absurdity becomes evident when calculating the total energy, which is the area under the curve, i.e. the integral over all wavelengths:

$U_\text{tot} \propto \int_0^\infty \frac{1}{\lambda^4}\,\dd\lambda = \infty$

The result is infinite. Taken literally, every lukewarm object — a lit heater, a human body — should instantly radiate an infinite amount of energy in the ultraviolet wavelengths and beyond, effectively annihilating the universe. This is the famous ultraviolet catastrophe, an expression coined by Paul Ehrenfest in 1911: the name highlights that the trouble explodes in the region of small $\lambda$ (the ultraviolet and beyond).

Experimentally, of course, nothing of the sort happens. The real curve rises, reaches a peak at $\lambda_\text{max}$ and then drops to zero exactly where the classical formula would diverge. The comparison between the two curves is merciless: they coincide in the infrared, but in the ultraviolet the classical prediction and the experimental data go in opposite directions. This was not a small numerical discrepancy to be corrected: classical physics was radically incompatible with observation.

The classical Rayleigh-Jeans prediction (red) diverges as $1/\lambda^4$ for $\lambda \to 0$: this is the ultraviolet catastrophe. The actually measured curve (blue, later explained by Planck) instead has a peak at $\lambda_\text{max}$ and collapses to zero towards short wavelengths.

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Topics: Quantum physics Concepts: Black body and Planck’s hypothesis

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