What is the reason for the shape of black-body radiation?

The shape of the electro-magnetic radiation spectrum of a blackbody is not trivial to understand. In fact, understanding this shape led to the birth of quantum mechanics at the beginning of the 20th century.

The Classical (pre quantum mechanics) picture:

Think of a blackbody as a spherical cavity in which photons are emitted and absorbed by the walls of the cavity.

Classically, the predicition is that the number of available states, or modes, for light-waves would go as 1/wavelength^4. The shorter the wavelength, the more waves 'fit in' to the cavity -- three factors of wavelength here, one for each spatial dimension. The shorter the wavelength, the more wave peaks (photons) are seen per unit time (fourth factor of wavelength).

This leads to what is called the "Rayleigh-Jeans" formula, and indeed the red (long-wavelength) side of a blackbody is perfectly described by this formula.

However, this leads to what was known in the last century as the `ultraviolet catastrophy:' there would be more and more photons at shorter and shorter wavelengths (and higher energy). This would lead to a blackbody emitting an INFINITE amount of energy. Something was wrong with the classical picture.

Planck (1905) asks what happens if energy is quantized?

The result, although not as intuitive as the above classical description, is that there are fewer states at shorter wavelengths (higher energies). This removes the `ultraviolet catastrophy;' the blackbody spectrum declines eventually at shorter wavelengths because this corresponds to higher energies where there are fewer states.

Now think of a blackbody spectrum as the most probable distribution of wavelengths (photons) for a perfect thermal emitter given its temperature.

A helpful analogy is to the velocity distribution of particles in a classical gas of some temperature. Temperature here is a measure of the mean velocity, or kinetic energy, of ths gas particles. For a perfect thermal emitter, the distribution is in wavelength instead of velocity, and the shape of the distribution is different. But what is the same is a concept of a statistical distribtuion with a characteristic shape and peak that is proportional to the temperature of the system.

Lecture 3 Lecture page Astro103 page

Last updated: Jan 12, 2001 Matthew A. Bershady