What is wrong with these ways of determining the mean occupation number?

Could anyone point out what went wrong in this argument?

Setup:

We have a system with 2 energy levels say with energies $0,e$ respectively. We consider the grand canonical ensemble for the system occupied by identical spinless fermions. Then there are clearly 4 feasible microstates --

1. no particles, $E=0$;
2. 1 particle in $E=0$ level, $E=0$;
3. 1 particle in $E=e$ level, $E=e$;
4. 2 particles one in each level, $E=e$

Clearly the corresponding grand canonical function is $$\Xi = 1+f+f\exp(-\beta e)+f^2\exp(-\beta e)$$ where $f\equiv \exp(\mu\beta)$.

Now I wish to find the mean occupation number of a state with energy $e$.

Method 1:

$$n(e)=1\times \text{probability of state (3)}+2\times \text{probability of state (4)}\\ = {f\exp(-\beta e)+2f^2\exp(-\beta e)\over \Xi}\\ ={1+2f\over 1+f}\cdot{1\over f^{-1}\exp(\beta e)+1}$$

BUT...

Method 2:

If I use the F-D distribution directly, shouldn't I get $$n(e) = {\text{degeneracy of energy level E=e}\over f^{-1}\exp(\beta e)+1}\\= 2\cdot{1 \over f^{-1}\exp(\beta e)+1}$$??

What went wrong?

Christie

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+1 concise question and nice argumentation beginning from basics – Stefan Bischof Apr 4 '13 at 15:56
in method 1 why did you do 2* probability of state 4(isnt it one microstate). and in method 2 why is there a degeneracy as you considered spinless identical fermions. – Prathyush Apr 4 '13 at 16:19
@Prathyush: I included the factor 2 because there are 2 particles in state (4). Is that wrong? – Christie Apr 4 '13 at 16:33
@StefanBischof: Thank you! :) – Christie Apr 4 '13 at 16:38
@Prathyush: Perhaps, you could give me a precise definition of "mean occupation number"? Perhaps I have understood it wrong... – Christie Apr 4 '13 at 16:47

There is a difference between energy levels (sometimes called states) and microstates (sometimes also called states).

You want to find how many particles are in energy level $e$, given a system with energy levels $0$ and $e$. Okay. Now, you did the right thing by considering the four distinguishable microstates, assigning likelihoods to them based on temperature, chemical potential, and the energies of the levels within each state.

But, when adding contributions in the numerator of Method 1, you are only supposed to count each microstate with a multiplicity given by how much it populates the energy level of interest. Microstates 1 and 2 have no particle in energy level $e$, so they are excluded. Microstate 3 has a single particle in energy level $e$, so it is counted once. Microstate 4 also has a single particle in energy level $e$, so it should only be counted once. The fact that it has another particle in the ground state doesn't matter - maybe it also describes a boson at energy level $e'$ on the far side of the Moon - that doesn't contribute to the counting of occupations of energy level $e$ by this species of fermion.

The only difference that ground state fermion in microstate 4 makes is in changing the chemical potential of the state, which you already accounted for with the $f^2$. It could also conceivably alter the Boltzmann factor for the state, but we have a normalization where the ground state has a Boltzmann factor of unity, so there is nothing more to consider.

So remove the 2 in method 1.

For Method 2, why the degeneracy factor? That would only be there if there were 2 distinct energy levels $e$. But your fermions are spinless - there is no other quantum degree of freedom for them to differ, so only 1 at most can ever occupy energy level $e$ in any microstate. In fact, you made use of this by not considering all the microstates that involve 2 particles in level $e$. So remove that 2 as well.

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THank you sooo much! It's absolutely clear now. You were right to have spotted my source of confusion. Thanks again! – Christie Apr 4 '13 at 17:12

In your Method 1, you forgot that the state (2) also has the occupation number $n=1$ (it has a vanishing energy but you're computing the mean occupation number, not mean energy), so it must be added to the numerator. This changes $1+2f$ in the numerator on the following line to $2+2f$. The simple fraction simplifies to $2$ and you get an agreement with Method 2.

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Thanks, Lubos! Would you mind explaining a bit about why I should include state (2) even though I am trying to find the mean occupation number for the energy level $e$ (given that state 2 has energy $0$)? – Christie Apr 4 '13 at 15:58
I think you didn't say that you're looking for the occupation number in the state $e$ only. If you are, then there are several errors. In Method 1, $2\times$ should be $1\times$, and in Method 2, the factor 2 should also be erased. At any rate $(1+f)/(1+f)$ cancel to $1$. – Luboš Motl Apr 4 '13 at 16:22
Thanks again, Lubos! Argh, I'm very confused now -- aren't there 2 particles therefore we should times 2? (I am very sorry about this, please be patient with me... :/ ) – Christie Apr 4 '13 at 16:40
Perhaps, you could give me a precise definition of "mean occupation number"? Perhaps I have understood it wrong...! – Christie Apr 4 '13 at 16:46
Thank you again, Lubos! – Christie Apr 6 '13 at 11:45