5.4 Ideal Bose gas

This function looks a bit like the Boltzmann factor but is everywhere larger than the latter. For small and large we again find that .

A popular example for this type of system is a ``gas'' of photons. A box with an internal coating of reflecting material in which a photon gas is held in thermodynamic equilibrium is often called a ``black body''. Originally this name refers to a grain of black material that might be placed in the box to allow for absorption and re-emission of photons, thus enabling energy exchange and equilibration. In actual fact the internal walls of the box are never perfectly reflecting, rendering the insertion of a black body grain unnecessary.

Since the total energy in the box is conserved and the photons may change
their energy upon absorption and reemission, the total number of
photons is **not conserved**. This is tantamount to assuming
.

To determine the energy spectrum of the photons we first calculate
the number of states within a small energy interval.
These states will then be populated according to equ. 5.12.
A simple geometric consideration - how many lattice points are lying
within a spherical shell - leads to

(5.13) |

(5.14) |

(5.15) |

(5.16) |

Providing an explanation for the experimentally measured spectrum of black body radiation was one of the decisive achievements in theoretical physics around 1900. Earlier attempts based on classical assumptions had failed, and it was Max Planck who, by the ad hoc assumption of a quantization of energy, could reproduce the correct shape of the spectrum. The subsequent efforts to understand the physical implications of that assumption eventually lead up to quantum physics.

2005-01-25