In this chapter, we explore one transformative experiment that helped pave the way to a new quantum theory: blackbody radiation and the ultraviolet catastrophe.   This experiment revealed a new quantum paradigm: the quantization of energy in matter.  

     In this section, we explore blackbody radiation.   A blackbody is a theoretical object that absorbs all incident frequencies of radiation, hence the name.  A blackbody that is in thermal equilibrium also emits all frequencies of radiation, though not equally.   The peak wavelength and energy density of emitted radiation depend on the blackbody's temperature.   Early classical attempts to explain blackbody radiation imagined that electromagnetic radiation emitted by the walls of the blackbody was caused by the oscillation of the electrons in constituent atoms. Implicit in this assumption was the idea that the electronic oscillators could have any arbitrary energy and that all oscillators contribute equally to the energy.   These ideas led to the following expression for the emitted energy density, known as the Rayleigh-Jeans Law:

$\mathrm{\ρ}\left(\mathrm{\ν}\, T\right)\=\frac{8 \mathrm{\π} k T}{c^3}\cdot {\mathrm{\nu }}^2$               (1)   Rayleigh-Jeans Law

Plot Rayleigh-Jeans Law for a blackbody at room temperature using the Maple input below:

What is wrong with the Rayleigh-Jeans law?

	Answer
	The energy density emitted increases as frequency increases, diverging to infinity!  So even at room temperature, a blackbody would be emitting UV-radiation and even higher energy radiation!

The 'ultraviolet catastrophe' was resolved when Max Planck proposed the quantization of energy, the revolutionary idea that the energy of each electric oscillator is limited to discrete values and cannot be varied arbitrarily. Furthermore, the permitted energies of an oscillator of a given frequency are integer multiples of a constant, h:

$E \= \mathrm{nh\ν}$  where n = 0, 1, 2, 3, ...        (2)  Quantization of energy

where the constant h was an unknown proportionality constant to give the proper units of energy. According to Planck's hypothesis, oscillators are only excited if they can acquire an energy of at least hn\027. Therefore, for the higher frequency oscillators, this energy is too large for the walls to supply, and they remain unexcited. Accounting for the quantization of energy leads to the Planck distribution:

$\mathrm{\ρ}\left(\mathrm{\ν}\, T\right)\=\frac{8 \mathrm{\π} h {\mathrm{\ν}}^3}{c^3} \cdot \frac{1}{\exp \left(\frac{\mathrm{h\ν}}{\mathrm{kT}}\right)-1}$               (3)   Planck's Distribution

Planck's distribution contained the unknown parameter, h.   A fit to experimental curves revealed h = 6.626 ×${10}^{-34}$, now known as Planck's constant! Using the Maple input below, plot Planck's distribution and vary T to see how the distribution depends on temperature.  

What do you notice about r as T increases?

	Answer
	As T increases, the energy density increases and becomes more broad, and the peak wavelength shifts to the blue, as expected!

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