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equilibrium situation for a quantum ideal gas in a harmonic trap which is a
harmonic potential containing a large number particles which do not interact with each
other except for instantaneous thermalizing collisions. This situation is of great
practical importance since many experimental studies of Bose Gas es are conducted
in such harmonic traps.

Using the results from either Maxwell-Boltzmann Statistics ,
Bose-Einstein Statistics or Fermi-Dirac Statistics we use the
Thomas-Fermi Approximation
and go to the limit of a very large trap, and express the degeneracy of the energy
states (''gi '') as a differential, and summations over states as
integrals. We will then be in a position to calculate the thermodynamic properties of
the gas using the Partition Function or the Grand Partition Function . Only the
case of massive particles will be considered, although the results can be extended to
massless particles as well, much as was done in the case of the ideal
Gas In A Box . More complete calculations will be left to separate articles, but
some simple examples will be given in this article.


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
'' {Link without Title} ''. The energy of a particular state
is given by:

:E=\hbar\omega\left(n_x+n_y+n_z+3/2 ight)~~~~~~~~n_i=0,1,2,\ldots

Suppose each set of quantum numbers specify ''f '' states where ''f '' 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 ''f=2 '', 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 ''n '', we can estimate the number of states with energy less than or equal to
''E '' from the above equation as

:g=f\, rac{n^3}{6}=f\, rac{(E/\hbar\omega)^3}{6}

which is just ''f '' 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 ''E '' and ''E+dE '' is therefore

:
dg= rac{1}{2}\,fn^2\,dn= rac{f}{(\hbar\omega\beta)^3}~ rac{1}{2}~
\beta^3 E^2\,dE


Notice that in using this continuum approximation, we have lost the ability to
characterize the low-energy states including the ground state where ''n_i=0 ''.
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
εi  is given by

:
N_i = rac{g_i}{\Phi}


where

with \beta=1/kT with ''k'' being Boltzmann's Constant ,
''T'' being Temperature , and μ being the Chemical Potential . Using the
continuum approximation, the number of particles ''dN'' with energy between ''E'' and ''E+dE'' is now written:

:dN= rac{dg}{\Phi}


THE ENERGY DISTRIBUTION FUNCTION


We are now in a position to determine some distribution functions for the "gas in a
harmonic trap" The distribution function for any variable A is PAdA 
and is equal to the fraction of particles which have values for ''A '' between
''A '' and ''A+dA ''

:P_A~dA = rac{dN}{N} = rac{dg}{N\Phi}

It follows that:

:\int_A P_A~dA = 1

Using these relationships we obtain the energy distribution function:

:
P_E~dE =
rac{1}{N}\,\left( rac{f}{(\hbar\omega\beta)^3} ight)~ rac{1}{2}
~ rac{\beta^3E^2}{\Phi}\,dE



SPECIFIC EXAMPLES


The following sections give an example of results for some specific cases.


Massive Maxwell-Boltzmann particles


For this case:

:\Phi=e^{\beta(E-\mu)}\,

Integrating the energy distribution function and solving for ''N '' gives

:N = rac{f}{(\hbar\omega\beta)^3}~e^{\beta\mu}

Substituting into the original energy distribution function gives

:P_E~dE = rac{\beta^3 E^2 e^{-\beta E}}{2}\,dE


Massive Bose-Einstein particles


For this case:

:\Phi=e^{\beta \epsilon}/z-1\,

where ''z '' is defined as

:z=e^{\beta\mu}\,

Integrating the energy distribution function and solving for ''N '' gives

:N = rac{f}{(\hbar\omega\beta)^3}~ extrm{Li}_3(z)

Where Lis(z) is the Polylogarithm function. The polylogarithm term must always be positive and real, which means its value will go from 0 to \zeta(3) as ''z'' goes from 0 to 1. As the temperature goes to zero, β will become larger and larger, until finally β will reach a critical value
\beta_c where ''z''=1 and

:N = rac{f}{(\hbar\omega\beta_c)^3}~\zeta(3)

The temperature at which β=βc  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:

:N= rac{g_0z}{1-z}+ rac{f}{(\hbar\omega\beta)^3}~ extrm{Li}_3(z)

where the added term is the number of particles in the ground state. (The ground state
energy has been ignored.) 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:

:\Phi=e^{\beta(E-\mu)}+1\,

Integrating the energy distribution function gives

:1= rac{f}{(\hbar\omega\beta)^3}~\left[- extrm{Li}_3(-z) ight]

Where again, Lis(z) is the Polylogarithm function. Further results can be found in the article on the ideal Fermi Gas .


REFERENCES


  • Huang, Kerson, "Statistical Mechanics", John Wiley and Sons, New York, 1967

  • A. Isihara, "Statistical Physics", Academic Press, New York, 1971

  • L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996

  • C. J. Pethick and H. Smith, "Bose-Einstein Condensation in Dilute Gases", Cambridge University Press, Cambridge, 2004