Partition function problems

[Lh0907]

Homework Statement

1. If the system, which has N identical particles, only has two possible energy states

E=0,e(e is an energy) ,what's the ensemble average of E?

2. Find the partition function which has two identical Fermion system if the energy states only have

E=0,e.

Homework Equations

I think what's different about partition function for identical Boson, Fermion and Boltmann stat particle?

The Attempt at a Solution

I made a solution but I'm not sure this is wrong or correct.

1. Z1 = 1 +exp(-e/kT)

Then, the total partition function is Z = (Z1)^N / (N!).

So Ensemble average of energy is <E> = (1/Z) * e * exp(-e/kt).

This is my solution.

Is this right or not?

2. Z1 = 1+exp(-e/kT)

Because of identical these Fermion -> Z = 1/2 * Z1^2

...But it should be changed due to property of Fermion.

The first identical particle can occupy E=0,e , so Z1 = 1+exp(-e/kT).

Then, if first particle occupy state which state energy is 0, Z2 can only exp(-e/kT).

Therefore partition function is (1+exp(-e/kT))*(-e/kT).

And another sitution is first particle occupy energy e state.

Then, another partition function is (1+exp(-e/kT)*1.

I thought this is connected by linearly

Final partition function is Z = 1/2 * (1/2*((1+exp(-e/kT))*(-e/kT)+ (1+exp(-e/kT)*1))

But I believe something is wrong this second problem.

What's the wrong?

 

[mark726]

I would like to provide some feedback on your solution attempt.

1. Your solution for the first problem seems to be correct. The partition function for a system with two energy states is indeed Z = 1 + exp(-e/kT). However, the expression for the ensemble average of energy should be <E> = (1/Z) * [0*exp(0) + e*exp(-e/kT)] = e/(1+exp(-e/kT)), as there are two possible energy states and the probability of each state is given by the Boltzmann factor.

2. For the second problem, the partition function for a system of identical fermions should take into account the Pauli exclusion principle, which states that no two identical fermions can occupy the same energy state. Therefore, the partition function for two identical fermions in two energy states would be Z = (1+exp(-e/kT))*(1+exp(-e/kT-1)), as the first fermion can occupy either energy state and the second fermion can occupy the remaining energy state. This can be extended to N identical fermions as Z = (1+exp(-e/kT))^N * (1+exp(-e/kT-1))^N / N!. The ensemble average of energy can then be calculated as <E> = (1/Z) * [0*exp(0) + e*exp(-e/kT) + 2e*exp(-2e/kT)] = 2e/(1+exp(-e/kT))^2, taking into account the possible combinations of energy states that the fermions can occupy.

Overall, your solution shows a good understanding of the concept of partition function and the differences between identical bosons, fermions, and Boltzmann particles. The only mistake was in the expression for the ensemble average of energy in the first problem. Keep up the good work!