The sun radiates energy. The earth absorbs a certain (tiny) fraction of that energy, which causes heating. The earth must lose energy in order to come to equilibrium. We conclude that the earth and other objects in the universe must emit energy in order to have a stable temperature. Then we must consider why the temperature of the earth is not the same as that of the sun. To explain this differene physicists in the 19th century noted that different objects have different emission wavelengths. This led to the Wien displacement law.
The emission spectra of stars give us information about the emission of all objects in the universe. Stars are convenient because they are very bright and we can measure the distribution of energy emitted at different wavelengths even using fairly simple devices. Therefore, scientists knew that different stars had different emission spectra and different maxima. The emission is known as "blackbody emission". This is a terrible name, at least for students since blackbodies are not black at all. The physicists surely knew what they meant by blackbody radiation, but let's just call it thermal radiation. The point is that star's have a thermal radiation spectrum that follows the Wien displacement law.
Failure of classical physics
The classical physics of radiation is described using Maxwell's equations as a starting point. These equations describe the relationship between electric and magnetic fields in space both for propgating waves and waves at interfaces. The equations can explain the phenomena that we observe in daily life, reflection, refraction and the propagation of light. However, when physicists attempted to explain the thermal radiation relationship using Maxwell's equations they arrived at an absurd result. It was known as the ultraviolet catastrophe.
Planck's solution to the ultraviolet catastrophe
The solution to this problem was proposed by Max Planck. He proposed that the light in a thermally emitting object was constrained to certain frequencies. Instead of all frequencies being possible, the frequencies were assumed to be an evenly spaced set with proportionality constant known as Planck's constant.
To determine the average energy of the light in the presence of a "ladder" of energy levels, Planck used a result derived by Ludwig Boltzmann. A few years earlier Boltzmann had exaplined thermodynamics using a novel statistical approach. One of the successes of Boltzmann's theory was the ability to calculate the energy of a system if one knew the energy levels in that sytem. On a side note, Boltzmann was ridiculed by many scientists in the day because his theory was based on the idea that there were energy levels. Before Planck's theory the idea of energy levels was not known. Boltzmann was a head of his time. Unfortunately, Boltzmann also had some tendency towards depression. He kllled himself in 1906, three years before he would have been vindicated and won the Nobel Prize with Planck. The moral of the story is "Never give up!".
Implications of the Planck law
- Planck's assumption places a restriction on the possible frequencies such that the energy E = nhn, where n is an integer and h is a constant. This fundamental assumption changes the nature of the problem in a fundamental way. Classically, the average energy is E = kBT. Based on a model that uses the evenly spaced set of levels one must use the averaging used by Boltzmann at low temperature. This value discussed in the section on the Partition function converts to the classical value at sufficiently high temperature.
- The Planck distribution is a bounded function with a maximum. That maximum shifts with temperature and follows the Wien displacement law. Thus, this experimental observation is satisfied by the Planck theory.
- The integrated area under the Planck curve increases with the fourth power of temperature in accord with the Stefan-Boltzmann law. This experimental observation is also explained by the Planck law.
- The fact that light is quantized implies that light has particle-like properties. The wave-particle duality is implied by the quantization of radiation.
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