Failure of Classical PhysicsThrowing all of this together (i.e. energy density is standing waves per volume times energy per standing wave), we get:
u(λ) = (8π / λ4) kTUnfortunately, the Rayleigh-Jeans formula fails horribly to predict the actual results of the experiments. Notice that the radiancy in this equation is inversely proportional to the fourth power of the wavelength, which indicates that at short wavelength (i.e. near 0), the radiancy will approach infinity. (The Rayleigh-Jeans formula is the purple curve in the graph to the right.)
R(λ) = (8π / λ4) kT (c / 4) (known as the Rayleigh-Jeans formula)
The data (the other three curves in the graph) actually show a maximum radiancy, and below the lambdamax at this point, the radiancy falls off, approaching 0 as lambda approaches 0.
This failure is called the ultraviolet catastrophe, and by 1900 it had created serious problems for classical physics because it called into question the basic concepts of thermodynamics and electromagnetics that were involved in reaching that equation. (At longer wavelengths, the Rayleigh-Jeans formula is closer to the observed data.)
Planck’s TheoryIn 1900, the German physicist Max Planck proposed a bold and innovative resolution to the ultraviolet catastrophe. He reasoned that the problem was that the formula predicted low-wavelength (and, therefore, high-frequency) radiancy much too high. Planck proposed that if there were a way to limit the high-frequency oscillations in the atoms, the corresponding radiancy of high-frequency (again, low-wavelength) waves would also be reduced, which would match the experimental results.
Planck suggested that an atom can absorb or reemit energy only in discrete bundles (quanta). If the energy of these quanta are proportional to the radiation frequency, then at large frequencies the energy would similarly become large. Since no standing wave could have an energy greater than kT, this put an effective cap on the high-frequency radiancy, thus solving the ultraviolet catastrophe.
Each oscillator could emit or absorb energy only in quantities that are integer multiples of the quanta of energy (epsilon):
E = n ε, where the number of quanta, n = 1, 2, 3, . . .The energy of each quanta is described by the frequency (ν):
ε = h νwhere h is a proportionality constant that came to be known as Planck’s constant. Using this reinterpretation of the nature of energy, Planck found the following (unattractive and scary) equation for the radiancy:
(c / 4)(8π / λ4)((hc / λ)(1 / (ehc/λ kT – 1)))The average energy kT is replaced by a relationship involving an inverse proportion of the natural exponential e, and Planck’s constant shows up in a couple of places. This correction to the equation, it turns out, fits the data perfectly, even if it isn’t as pretty as the Rayleigh-Jeans formula.