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# 5.1 Ideal quantum gas: Method of the most probable distribution

We consider a system of independent particles in a cubic box with ideally elastic walls. In the spririt of the kinetic theory of dilute gases we explore the state space for the individual particles, which Boltzmann dubbed -space. In the quantum case this space is spanned by the quantum numbers pertaining to the momentum eigenstates having eigenvalues and energies .

Now we batch together all states having energies in an interval . The number of states in such a cell'' is named . The values of the are not important; they should only be large enough to allow the application of Stirling's formula.

As before we try to answer the question how the particles should best be distributed over the cells. To do so we change the notation from the one used in Section 2.2, in that we denote the number of particles in cell by . The reason for using is that '' is reserved for the quantum numbers.

A specific distribution of the particles to the cells is more probable if its multiplicity is larger, meaning that we can allot the particles in more different ways to the states in each cell - always keeping in mind the Fermi or Bose rules:

 (5.1)

To compare: the multiplicity given in Section 2.2, pertaining to a classical distribution (see equ. 2.20) would in our present notation read .

The distribution having the largest multiplicity may again be determined by Lagrange variation with the conditions and :

 (5.2)

Writing we find for
 (5.3)

This is the most probable distribution of the particles upon the cells. Since denotes the number of states in cell , we have for the average population number of each state
 (5.4)

It is easy to interpret the Lagrange parameters and . As in Section 2.2 one compares the consequences of the population densities given above to empirical/thermodynamical facts, finding that is related to temperature as , and that is identical to the fugacity.

For a better understanding of this derivation, let us interpret its premises as a set of rules in a game of fortune, as we have done in Chapter 2. By running the applet EFRoulette we may indeed play that game - for Fermi particles at least - and compare its outcome with the result just given.

These are the rules:

For non-interacting particles in a square box the -plane is spanned by integers ; each quantum state is represented by a point. A specific state of a system of fermions is represented by a set of inhabited points on that plane.

To find the average (and also most probable!) distribution of particles on states,

- assign particles randomly to the states on -plane

- make sure that the sum of the particle energies equals the given system energy,
AND
- discard all trials in which a state is inhabited by more than one particle

- determine the mean number of particles in each state; sort the result according to the state energies

 Applet LBRoulette: Start Playing the Fermi-Dirac game. [Code: EFRoulette]

Next: 5.2 Ideal quantum gas: Up: 5. Statistical Quantum Mechanics Previous: 5. Statistical Quantum Mechanics
Franz Vesely
2005-01-25