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Consider a gas consisting of identical noninteracting particles
occupying volume and in thermal equilibrium at temperature .
Let us label the possible quantum states of a single particle by (or ).
Let the energy of a particle in state be denoted .
Let the number of particles in state be written . Finally,
let us label the possible quantum states of the whole gas by .
The particles are assumed to be noninteracting, so the
total energy of the gas in state , where there are particles
in quantum state , etc., is simply

(579) 
where the sum extends over all possible quantum states . Furthermore,
since the total number of particles in the gas is known to be , we must
have

(580) 
In order to calculate the thermodynamic properties of the gas (i.e.,
its internal energy or its entropy), it is necessary to
calculate its partition function,

(581) 
Here, the sum is over all possible states of the whole gas:
i.e., over all the various possible values of the
numbers
.
Now,
is
the relative probability of finding the gas in a particular state in which
there are particles in state 1, particles in state 2, etc.
Thus, the mean number of particles in quantum state can be written

(582) 
A comparison of Eqs. (581) and (582) yields the result

(583) 
Here,
.
Next: FermiDirac statistics
Up: Quantum statistics
Previous: An illustrative example
Richard Fitzpatrick
20060202