Ultraviolet, not ultraviolence
In some of the past BMC posts, I have blogged about how statistical mechanics, in the 19th century, came perilously close to uncovering quantum mechanics early. A number of “problems” with statistical mechanics arose due to the classical treatment used. One of the most serious was the Ultraviolet Catastrophe. This problem was not easily solved with the hand-wavey pseudo-quantum explanations used in some previous cases. It took a full-on, quantized description of electromagnetism, and helped usher in quantum mechanics to being the foremost theory in subatomic physics.
Black body Radiation
The catastrophe arises from considering a phenomena called black body radiation. A black body is a perfect absorber, and a perfect emitter of radiation. What that means, is that all the light, x-rays, microwaves, etc. (all forms of electromagnetic radiation) striking the object are completely absorbed (increasing its temperature), and that it is a perfect radiator (it glows because of its temperature). The glow it gives off increases in frequency as the object heats up (which is why a hot steel will go from red to yellow to white as it heats). A black body can be used as an approximation for most physical objects, from stars to people, but is one of the many non-physical objects used in physics like massless strings and frictionless planes. Perhaps Hotblack Desiato could make one, but we haven’t yet.
The Ultraviolet Catastrophe
The problem arose when trying to determine how much power was radiated by the black body. When physicists in the 19th century attempted to calculate this, they turned to two tools: classical harmonic oscillators (things like vibrations on a string), and the equipartition theorem. The classical harmonic oscillator theory of radiation told them that the number of oscillation modes in a 3D box is proportional to the frequency of the wave squared, and that the power of a wave was dependent on the frequency of it. The equipartition theorem told them that each mode of vibration was a degree of freedom, and that it stored a fixed amount of energy, dependent on the temperature. BIG SCIENCY WARNING BELLS WENT OFF. These two things give you that as you increase the vibration mode, you get a massive increase in degrees of freedom, and therefore energy, all the way up to infinity. These calculations told them that any object above absolute zero emits an infinite amount of radiation, most of it in ultra-high energy gamma rays (ultraviolet was the whiz-bang high energy light of the time, hence the name).
Planck Saves the Day
Nice 'stache, Max Planck!
The solution to this catastrophe came from none other than Max Planck. He modified the idea of using classical harmonic oscillators as a model for radiation by introducing quantized emission. In other words, the photon: a discrete “packet” of EM radiation. These packets have energy proportional to their frequency, which means that as you approach infinite frequency, their energy expands to infinity as well. This dropped the number of available modes in the cavity for higher frequencies, causing the emission to go to zero at very high frequencies (rather than ballooning to infinity). The result of this modification was a change in how radiation was viewed (which, along with Einstein’s photoelectric effect helped cement the idea of photons) from a classical oscillator to a gas of Bosons (non-excluding, indistinguishable particles). The result of Planck’s work was Planck’s Law of black body radiation, which accurately describes the spectrum of hot objects, allowing us to easily gauge the temperature of objects by examining their radiation.
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy (i.e., the sum of emissions in all frequency ranges), it can be shown that a blackbody would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed.
The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 105 GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum. Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.
The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of .
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. And, 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 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 is unlimited as higher and higher frequencies are considered: this is clearly unphysical as the total radiated power of a cavity is not observed to be infinite, a point that was made independently by Einstein and by Lord Rayleigh and Sir James Jeans in 1905.
Max Planck derived the correct form for the intensity spectral distribution function by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta, of energy: , where h is Planck's constant. Planck's assumptions led to the correct form of the spectral distribution functions: . Albert Einstein solved the problem by postulating that Planck's quanta were real physical particles—what we now call photons, not just a mathematical fiction. He modified statistical mechanics in the style of Boltzmann to an ensemble of photons. Einstein's photon had an energy proportional to its frequency and also explained an unpublished law of Stokes and the photoelectric effect. This published postulate was specifically cited by the Nobel Prize in Physics committee in their decision to award the prize for 1921 to Einstein.
- ^McQuarrie, Donald A.; Simon, John D. (1997). Physical chemistry: a molecular approach (rev. ed.). Sausalito, Calif.: Univ. Science Books. ISBN 978-0-935702-99-6.
- ^Mason, Stephen F. (1962). A History of the Sciences. Collier Books. p. 550.
- ^Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press.
- ^"The Nobel Prize in Physics: 1921". Nobelprize.org. Nobel Media AB. 2017. Retrieved December 13, 2017.