The results of the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap, which is a harmonic potential containing a large number of particles that do not interact with each other except for instantaneous thermalizing collisions. This situation is of great practical importance since many experimental studies of Bose gases are conducted in such harmonic traps.
Contents
- Thomas Fermi approximation for the degeneracy of states
- The energy distribution function
- Specific examples
- Massive MaxwellBoltzmann particles
- Massive BoseEinstein particles
- Massive FermiDirac particles eg electrons in a metal
- References
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 (
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
Suppose each set of quantum numbers specify
which is just
Notice that in using this continuum approximation, we have lost the ability to characterize the low-energy states, including the ground state where
Without using the continuum approximation, the number of particles with energy
where
with
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
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
Substituting into the original energy distribution function gives:
Massive Bose–Einstein particles
For this case:
where
Integrating the energy distribution function and solving for
Where
The temperature at which
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:
Integrating the energy distribution function gives:
where again,