Thomas–Fermi approximation for the degeneracy of states
For massive particles in a harmonic well, the states of the particle are enumerated by a set of quantum numbers. The energy of a particular state is given by: Suppose each set of quantum numbers specify states where is the number of internal degrees of freedom of the particle that can be altered by collision. For example, a spin-1/2 particle would have, one for each spin state. We can think of each possible state of a particle as a point on a 3-dimensional grid of positive integers. The Thomas–Fermi approximation assumes that the quantum numbers are so large that they may be considered to be a continuum. For large values of, we can estimate the number of states with energy less than or equal to from the above equation as: which is just times the volume of the tetrahedron formed by the plane described by the energy equation and the bounding planes of the positive octant. The number of states with energy between and is therefore: Notice that in using this continuum approximation, we have lost the ability to characterize the low-energy states, including the ground state where. For most cases this will not be a problem, but when considering Bose–Einstein condensation, in which a large portion of the gas is in or near the ground state, we will need to recover the ability to deal with low energy states. Without using the continuum approximation, the number of particles with energy is given by: where with, with being Boltzmann's constant, being temperature, and being the chemical potential. Using the continuum approximation, the number of particles with energy between and is now written:
We are now in a position to determine some distribution functions for the "gas in a harmonic trap." The distribution function for any variable is and is equal to the fraction of particles which have values for between and : It follows that: Using these relationships we obtain the energy distribution function:
Specific examples
The following sections give an example of results for some specific cases.
Massive Maxwell–Boltzmann particles
For this case: Integrating the energy distribution function and solving for gives: Substituting into the original energy distribution function gives:
Massive Bose–Einstein particles
For this case: where is defined as: Integrating the energy distribution function and solving for gives: Where is the polylogarithm function. The polylogarithm term must always be positive and real, which means its value will go from 0 to as goes from 0 to 1. As the temperature goes to zero, will become larger and larger, until finally will reach a critical value, where and The temperature at which is the critical temperature at which a Bose–Einstein condensate begins to form. The problem is, as mentioned above, the ground state has been ignored in the continuum approximation. It turns out that the above expression expresses the number of bosons in excited states rather well, and so we may write: where the added term is the number of particles in the ground state. This equation will hold down to zero temperature. Further results can be found in the article on the ideal Bose gas.
Massive Fermi–Dirac particles (e.g. electrons in a metal)
For this case: Integrating the energy distribution function gives: where again, is the polylogarithm function. Further results can be found in the article on the ideal Fermi gas.
