For bosons the mean population number of a state is
This function looks a bit like the Boltzmann factor
but is everywhere larger than the latter.
For small and large we again find that
Mean population number of states in a Bose-Einstein system with
For comparison we include a graph of the classical density
PHOTONS IN A BOX
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
where is the momentum pertaining to the energy
The total number of photons in the system is thus
and accordingly the number of photons in a frequency interval
The amount of energy carried by these photons is the
spectral density of black body radiation; it is given by
For pressure and internal energy of the photon gas we find
- in contrast to the classical and Fermi cases - the relation
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.