The Ultraviolet Catastophe
Planck's Law was first derived empiricaly to bridge the|
gap between the Rayleigh-Jeans Law which worked at low
frequencies and Wien's Law that worked at high
frequencies. He later created a theoretical foundation
for the law based on the quantization of energy modes.
Why the Classical Model Blows Up at Higher Frequencies In order to calculate the density of oscillation modes as a function of temperature the Rayleigh-Jeans formulation considered standing electromagnetic waves within a cubic oven. As a boundary condition the parallel components of each electromagnetic wave mode was forced to zero at the oven walls. By counting the number of possible oscillation modes within an increment of frequency it was predicted that the modal density per unit frequency, per unit volume, would be:
Using the Equipartition of Energy principle we would predict that the blackbody radiation intensity would also vary by this factor. As seen in the graph above, the classical model predicts that radiation intensity will increase indefinitely in proportion to the square of the freqency. This prediction works well for lower frequencies but not at all for higher frequencies.
To understand the reasonableness of the nonconvergence of the modal density curve, we can look at the simpler one dimensional case where the standing waves are set between two walls of an oven located at x=0 and x=a. Our boundary conditions stipulate that modal waves go to zero at the walls. Possible solutions are therefore of the form:
So we can see even that even for the one dimensional model, the classical formulation predicts that the modal frequency density, and hence the blackbody intensity distribution, will not converge to zero for large frequencies and will not provide us with a finite value for the total radiated energy.