Quantum Mechanics: The Ultraviolet Catastrophe

File:Light shining1.JPG
Zouavman Le Zouave / CC BY-SA (https://creativecommons.org/licenses/by-sa/3.0); https://commons.wikimedia.org/wiki/File:Light_shining1.JPG

The Rayleigh-Jeans law, based on classical physics, gives the spectral radiance (linked to intensity) of blackbody electromagnetic radiation as a function of wavelength:

Bv(T) is spectral radiance; kB is the Boltzmann constant; T is temperature; c is the speed of light; λ is wavelength. [1]

Blackbodies are defined as ‘[idealised] objects that perfectly absorb and then re-emit radiation’ [2]. The equation shows that increasing the wavelength should cause the spectral radiance of the emitted radiation to decrease and tend to zero whilst decreasing wavelength causes it to increase, tending to infinity.


This graph [3] shows what the Rayleigh-Jeans law predicts (labelled ‘Classical theory’) and what actually happens at different temperatures (the blue, green and red lines). Although the law seems to coincide with experimental data at higher wavelengths, they contrast significantly at lower wavelengths. Evidence reveals that at low wavelengths (high frequencies), spectral radiance decreases. This problem is known as the ultraviolet catastrophe, simply because the disparity between the law and experiments becomes most apparent in the ultraviolet frequency range and beyond.

The issue with the formula is the underlying classical physics. Rayleigh and Jeans incorporated the Equipartition theorem of classical statistical mechanics which states that at equilibrium, ‘energy is evenly spread between all possible energy states.’ [4] In classical physics, particles can vibrate with any amount of energy. This meant that there can be infinitely many energy states as you can continuously divide their value: there is no smallest energy. When trying to mathematically redistribute heat energy between these energy states, 'way too much energy got packed into the countless very tiny energy states at high frequencies.' [4] This caused the spectral radiance at high frequencies to tend to infinity.

In order to solve this major problem, German physicist, Max Planck, used a mathematical trick: he suggested that the energy of the oscillators should be quantised. This means that energy can only be emitted and absorbed in discrete packets known as ‘quanta.’ The energy of these quanta is determined by the equation: E = hf where E is energy, h is Planck’s constant and f is the frequency of the oscillations. With this move, Planck solved the ultraviolet catastrophe as it limited the amount of energy that could be contained within the high frequency vibrations.

By solving the ultraviolet catastrophe, Planck sparked the quantum revolution. Later on, Einstein posited that these ‘quanta’ were real physical entities leading to the hypothesis of photons.

Sources:
1. https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Jeans_law#Historical_development
2. The Quantum Universe by Brian Cox and Jeff Forshaw
3. https://commons.wikimedia.org/wiki/File:Black_body.svg
4. https://www.youtube.com/watch?v=tQSbms5MDvY
https://en.wikipedia.org/wiki/Ultraviolet_catastrophe


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