Ultraviolet catastrophe

The phrase refers to the fact that the empirically derived Rayleigh–Jeans law, which accurately predicted experimental results at large wavelengths, failed to do so for short wavelengths. (See the image for further elaboration.) As the theory diverged from empirical observations when these frequencies reached the ultraviolet region of the electromagnetic spectrum, there was a problem. This problem was later found to be due to a property of quanta as proposed by Max Planck: There could be no fraction of a discrete energy package already carrying minimal energy.

Since the first use of this term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.

Problem

The Rayleigh-Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength ${\displaystyle \lambda }$, it is: $${\displaystyle B_{\lambda }(T)=\frac{2c{k}_{\mathrm{B}}T}{{\lambda }^4},}$$ where ${\displaystyle B_{\lambda }}$ is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength; ${\displaystyle c}$ is the speed of light; ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant; and ${\displaystyle T}$ is the temperature in kelvins. For frequency ${\displaystyle \nu }$, the expression is instead $${\displaystyle B_{\nu }(T)=\frac{2{\nu }^2{k}_{\mathrm{B}}T}{c^2}.}$$

This formula is obtained from the equipartition theorem of classical statistical mechanics which states that each one of all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of ${\displaystyle k_{\mathrm{B}}T}$.

The "ultraviolet catastrophe" is the expression of the fact that the formula misbehaves at higher frequencies; it predicts infinite energy emission because ${\displaystyle B_{\nu }(T)\to \mathrm{\infty }}$ as ${\displaystyle \nu \to \mathrm{\infty }}$.

An example, from Mason's A History of the Sciences, illustrates multi-mode vibration via a piece of string. As a natural vibrator, the string will oscillate with specific modes (the standing waves of a string in harmonic resonance), dependent on the length of the string. In classical physics, a radiator of energy will act as a natural vibrator. Since each mode will have the same energy, most of the energy in a natural vibrator will be in the smaller wavelengths and higher frequencies, where most of the modes are.

According to classical electromagnetism, the number of electromagnetic modes in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency. This implies that the radiated power per unit frequency should be proportional to frequency squared. Thus, both the power at a given frequency and the total radiated power is unlimited as higher and higher frequencies are considered: this is unphysical, as the total radiated power of a cavity is not observed to be infinite, a point that was made independently by Einstein, Lord Rayleigh, and Sir James Jeans in 1905.