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Planck's Law

As the blackbody temperature rises the total 
energy emitted increases and the frequency of 
the peak emissions increases. The surface of 
the sun is about 5700 deg K and emits with a 
radiation curve which peaks in the centre of the 
visible portion of the electomagnetic spectrum.
As the blackbody temperature rises the total
energy emitted increases and the frequency of
the peak emissions increases. The surface of
the sun is about 5700 deg K and emits with a
radiation curve which peaks in the centre of the
visible portion of the electomagnetic spectrum.

How Quantum Mechanics Resolves the Ultraviolet Catastophe



In order to explain the variation between the classical prediction for blackbody emissions and the experimental observations, Planck devised an empirical formula which could be used to accurately predict blackbody radiation curves for a wide range of temperatures,

where h and k are constants, T is temperature, f is frequency and c is the speed of light,

He then broke the expression into two factors. The first being the modal density devised for use with the Raliegh-Jean relation and a second residual term which must be interpreted as the average energy per mode as a function of frequency and temperature.

From here we see that the average energy per mode is not kT as it was in the Raliegh-Jean relation but rather:

This factor reduces nicely to kT matching the Raliegh-Jean relation for low frequencies, but for high frequencies it converges to zero as required to match experimental results and to avoid the 'ultraviolet catastophe.'

Planck showed that this expression for average energy per mode could be explained if nature in fact imposed a restriction on oscillation modes such that they could only take on energies which were integer multiples of hf,

and if the probability of entering into a specific mode was a function of the energy level of that mode.

This mechanism would explain why higher energy modes were more difficult to fill since we can see that the probability of entering into the first energy level for a specific mode, will become diminishingly small as the frequency factor hf becomes large relative to kT.

This was the first time that the quantization of energy was effectively proposed as a physical mechanism to explain atomic level phenomena. As such, this could be seen as the birth of quantum mechanics. In 1905 Einstein adopted Planck's concept as a mechanism to explain the photoelectric effect, and so quantum mechanics was launched upon the world.

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