Ultraviolet catastrophe

The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. As observation showed this to be clearly false, it was one of the first clear indications of problems with classical physics. The solution to this problem led to the development of an early form of quantum mechanics.

The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, although the concept goes back to 1905; the word "ultraviolet" refers to the fact that the problem appears in the short wavelength region of the electromagnetic spectrum. Since the first appearance of the term, it has also been used for other predictions of a similar nature, e.g. in quantum electrodynamics (also used in those cases: ultraviolet divergence).

The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all modes (degrees of freedom) of a system at equilibrium have an average energy of <math>kT/2<math>. 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 therefore implies that the radiated power per unit frequency should follow the Rayleigh-Jeans law, and be proportional to frequency squared. Thus, both the power at a given frequency and the total radiated power go to infinity as higher and higher frequencies are considered: this is clearly an impossibility, a point that was made independently by Einstein and by Lord Rayleigh and Sir James Jeans in the year 1905. Einstein pointed out that the difficulty could be avoided by making use of a hypothesis put forward five years earlier by Max Planck. Planck had postulated that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency (as given by Planck's law). This has the effect of reducing the number of possible modes with a given energy at high frequencies in the cavity described above, and thus the average energy at those frequencies by application of the equipartition theorem. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite. The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck's law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck's constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism.

Many popular histories of physics, as well as a number of textbooks, present an incorrect version of the history of the Ultraviolet Catastrophe. In this version, the "catastrophe" was first noticed by Planck, who developed his formula in response. In fact Planck never concerned himself with this aspect of the problem, because he did not believe that the equipartition theorem was fundamental - his motivation for introducing "quanta" was entirely different. It was several years later that physicists realized that Planck's law resolved a fundamental crisis of classical physics.

References

• Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. (W. H. Freeman and Company: New York, 1980). See chapter 4.
• Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum Mechanics: Volume One (Hermann: Paris, 1977), pp. 624 —626.
• Kragh, Helge (2000) Max Planck: The reluctant revolutionary (http://www.physicsweb.org/articles/world/13/12/8/1) Physics World December 2000fr:Catastrophe ultraviolette