UV Catastrophe (Black-Body Radiation)

To begin an overview of the evolution of Quantum Mechanics, one must first examine its birthplace, i.e. the black body radiation problem. It is simple to understand that emission of radiation from an object occurs for all temperatures greater than absolute zero. As the temperature of the object rises the energy concentration of the emitted radiation (the spectral distribution) shifts away from the long wavelength, i.e. infrared regions, to the shorter wavelength regions, including the visible spectrum and finally the UV and X-ray regions. Coherently, the total power radiated increases with temperature.

Imagine a perfect absorber cavity (i.e. it absorbs all radiation at all wavelengths, so that its spectral radiance only depends on temperature). From Kirchoff's law it follows that such a body would not only be a perfect absorber, but also a perfect emitter of radiation. "Blackbody" is a kind of material, not only it absorbs all of the radiation when the radiation falls on it, but also it seems black. This emission is called the black body radiation. Lord Rayleigh (John William Strutt) and Sir James Jeans applied classical physics and assumed that the radiation in this perfect absorber could be represented by standing waves. Although the Rayleigh-Jeans result does approach the experimentally recorded values for large values of wavelength, the trend line vastly differs as the wavelength is allowed to tend towards zero. The result predicts that the spectral intensity will increase quadratically with increasing frequency, and would diverge to infinite energy as the wavelength went to zero. For short wavelengths, this became known as the so called "Ultraviolet Catastrophe." This black body radiation experiment shows an important failure of classical mechanics. The Rayleigh-Jeans law is as follows:

$u_{\text{RJ}}(\nu ,T)={\frac {8\pi \nu ^{2}}{c^{3}}}~k_{\text{B}}T$

where $c\!$ is the speed of light, $k_{\text{B}}\!$ is Boltzmann's constant and $T\!$ is the temperature in Kelvins. ( There is a following relation between $u(\nu ,T)\!$ and $u(\lambda ,T)\!$:

${\begin{aligned}u(\nu ,T)&=u(\lambda ,T)\left|{\frac {d\lambda }{d\nu }}\right|\\&={\frac {c}{\nu ^{2}}}u(\lambda ,T).\end{aligned}}$

Based on a thermodynamic argument, Wien noted that under adiabatic expansion, the energy of a mode of light, the frequency of the mode, and the total temperature of the light change together in the same way, so that their ratios are constant. This implies that in each mode at thermal equilibrium, the adiabatic invariant energy/frequency should only be a function of the adiabatic invariant frequency/temperature. The Wein law is:

$u_{\text{Wien}}(\nu ,T)=\nu ^{3}g\left({\frac {\nu }{T}}\right)$

and Wein predicted that $g(\nu /T)=Ce^{h\nu /k_{\text{B}}T}\!$.

In 1900, Max Planck offered a successful explanation for black body radiation. He too postulated that the radiation was due to oscillations of the electron, but the difference between his assumption and Rayleigh's was that he argued that the possible energies of an oscillator were not continuous. He proposed that the energy of an oscillator would be proportional to a constant of the frequency.

$E=h\nu =\hbar \omega$

Here $E\!$ is energy, h is the Planck constant ($h=6.626*10^{-34}{\text{Joule-seconds}}\!$ ) and $\nu \!$ is the frequency of the oscillator. With the concept of energy being discrete in mind, the result is that Planck's calculation avoided the UV catastrophe, and instead the energy approached zero as the frequency tends to infinity increased. Planck's law of black body radiation is as follows:

$u_{(}\nu ,T)={\frac {8\pi h}{c^{3}}}{\frac {\nu ^{3}}{e^{\frac {h\nu }{k_{\text{B}}T}}-1}}$

When $\nu \to 0$ and $\nu \to \infty$, we can easily get Rayleigh-Jeans formula and Wien formula.

Before leaving the subject of Black Body Radiation it is important to look at one fundamental realization that has come out of the mathematics. In 1964, A. Penzias and R. Wilson discovered a radio signal of suspected cosmic origin, with an intensity corresponding to approximately 3 K. Upon application of Planck's theorem for said radiation, it soon became evident that the spectrum seen corresponded to that of a black body at 3 K, and since this radiation was incident on Earth evenly from all directions, space itself was deemed to be the emitting black body. This cosmic background radiation gave credence to the Big Bang theory, and upon analysis of an expanding system, allowed for proof that Planck's theorem holds for black bodies of changing size. The results of this particular proof even allow for a fair estimation into the rate of expansion of the universe since the time the black body radiation was emitted.