Editing Gas in a harmonic trap (section)

== 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 <math>[n_x,n_y,n_z]</math>. The energy of a particular state is given by:
: <math>E=\hbar\omega\left(n_x+n_y+n_z+3/2\right)~~~~~~~~n_i=0,1,2,\ldots</math>

Suppose each set of quantum numbers specify <math>f</math> states where <math>f</math> 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 <math>f = 2</math>, 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 <math>n</math>, we can estimate the number of states with energy less than or equal to <math>E</math> from the above equation as:
: <math>g=f\,\frac{n^3}{6}=f\,\frac{(E/\hbar\omega)^3}{6}</math>
which is just <math>f</math> 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 <math>E</math> and <math>E + dE</math> is therefore:
: <math>dg=\frac{1}{2}\,fn^2\,dn=\frac{f}{(\hbar\omega\beta)^3}~\frac{1}{2}~\beta^3 E^2\,dE</math>

Notice that in using this continuum approximation, we have lost the ability to characterize the low-energy states, including the ground state where <math>n_{i} = 0</math>. 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 <math>\epsilon_{i}</math> is given by:
: <math>N_i = \frac{g_i}{\Phi}</math>
where
: {|
|-
|<math>\Phi=e^{\beta(\epsilon_i-\mu)}</math>
|&nbsp;&nbsp;&nbsp;for particles obeying [[Maxwell–Boltzmann statistics]]
|-
|<math>\Phi=e^{\beta(\epsilon_i-\mu)}-1</math>
|&nbsp;&nbsp;&nbsp;for particles obeying [[Bose–Einstein statistics]]
|-
|<math>\Phi=e^{\beta(\epsilon_i-\mu)}+1</math>
|&nbsp;&nbsp;&nbsp;for particles obeying [[Fermi–Dirac statistics]]
|}

with <math>\beta=1/kT</math>, with <math>k</math> being the [[Boltzmann constant]], <math>T</math> being [[temperature]], and <math>\mu</math> being the [[chemical potential]]. Using the continuum approximation, the number of particles <math>dN</math> with energy between <math>E</math> and <math>E + dE</math> is now written:
: <math>dN= \frac{dg}{\Phi}</math>