# Partition functions of boxes containing bosons or fermions

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Suppose you have a "box" in which each particle may occupy any of 10 single-particle states. For simplicity, assume that each of these states has energy zero.

(a) What is the partition function of this system if the box contains only one particle?

(b) What is the partition function of this system if the box contains two distinguishable particles?

(c) What is the partition function if the box contains two identical bosons?

(d) What is the partition function if the box contains two identical fermions?

(e) What would be the partition function of this system according to equation Z = (1/N!)Z_1^N?

(f) What is the probability of finding both particles in the same single-particle state, for the three cases of distinguishable particles, identical bosons, and identical fermions?

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The partition function is the summation:

Z = Sum_{r} Exp[-beta E_{r}] (1)

where r denotes the states of the system.

If you have two particles then r must specify the states of the two particle system and E_{r} is the energy of the two particle system r. In case of non-interacting particles the energy of a multi-particle state is the sum of the energies of the individual particles.

In this problem all the energies of the single particle states are zero, and that means that the term Exp[-beta E_{r}] is always 1. This means we are summing 1 over all possible states, so we are just counting the number of possible states the system can be in!

Problem a) If we have only one ...

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