More on the grand canonical ensemble

We have previously derived a general expression for the grand canonical partition function for a system of independent identical particles

where the product over ? is actually the product over the single-particle eigenstates. Each eigenstate must be counted separately, even when several eigenstates have the same energy (also denoted by ?). The sum over n is only over allowed values of n. For bosons, an arbitrary number of particles can be in the same state, so n can take on any non-negative integer value, n = {0, 1, 2,...}. For fermions, no more than one particle can be in a given state, so n can only take on the values 0 and 1, n = {0, 1}. Do each of the following for both fermions and bosons

1. Carry out the sum in the grand canonical partition function explicitly.

2. Using the result from the previous question, calculate the average number of particles in terms of β and μ.

3. Calculate the average number of particles in an eigenstate with energy ?, denoted by ?>.

4. Express the average number of particles in terms of a sum over n.

5. Calculate the average energy in terms of a sum over ?>.