General Concepts About Fermi-Dirac Distribution

[MartinCort]

Hello!
Thanks for your time reading my questions.
When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions.
1. The Pauli's exclusion principle says that: Two fermions can't occupy the same quantum states.
2. The Fermi-Dirac distribution tells how many electrons there are in one quantum state with Energy E_i

Is there any possibility that we find a system, which has 2 degeneracy(for example) satisfying both of the requirements？
If so how do we interpret the Fermi-Dirac distribution in this case, because we know when E=E_i, there are two particles, but from the Fermi-Dirac distribution, the average number will be 1?

 

[PeterDonis]

The Fermi-Dirac distribution tells how many electrons there are in one quantum state with Energy E_i

No. The Fermi-Dirac distribution tells how many electrons there are (more precisely, the expectation value of electron number) when aggregated over all of the quantum states that have energy $E_i$. There might well be more than one such state.

we know when E=E_i, there are two particles

No, you don't. You know there are two states with energy $E_i$ (because you stipulated that in your hypothetical), but you don't know that there are electrons occupying both of those states.

 

[MartinCort]

No. The Fermi-Dirac distribution tells how many electrons there are (more precisely, the expectation value of electron number) when aggregated over all of the quantum states that have energy $E_i$. There might well be more than one such state.
No, you don't. You know there are two states with energy $E_i$ (because you stipulated that in your hypothetical), but you don't know that there are electrons occupying both of those states.

Hello Peter
Thanks for your explanation!

Can I understand this as follow?

It is possible to have two states A and B with the same energy, but we can not say that there is an electron in state A while there is another electron in state B. So the number of electrons will still be one which is consistent with the Fermi-Dirac distribution.

However, Would you mind elucidating why it is not allowed?

Thanks!

 

[PeterDonis]

It is possible to have two states A and B with the same energy, but we can not say that there is an electron in state A while there is another electron in state B.

No, that's not correct. The Pauli exclusion principle only says you can't have two fermions in the same state--so there can't be two electrons both in state A, or two electrons both in state B. But there is nothing stopping one electron being in state A and another electron in state B.

So the number of electrons will still be one which is consistent with the Fermi-Dirac distribution.

Why do you think the number of electrons has to be one if there are two states, A and B?

 

[Demystifier]

When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions.
(1) The Pauli's exclusion principle says that: Two fermions can't occupy the same quantum states.
(2) The Fermi-Dirac distribution tells how many electrons there are in one quantum state with Energy E_i

According to (1), the number of particles in a given state is either 0 or 1. But statistical physics assigns probabilities to those two possibilities, meaning that the average number of particles in a given state can be any real number in the interval [0,1]. According to (2), this average number is something like
$$\frac{1}{e^{\beta E}+1}$$
The crucial thing here is that we have the term +1 (not -1 as in the Bose-Einstein distribution), which provides that this number indeed cannot be larger than 1.

 

[PeterDonis]

this average number is

Note that this formula assumes no degeneracy. If there is degeneracy, the formula you give has to be multiplied by the degree of degeneracy to give the expectation value of particle number for energy $E$.

 

[Demystifier]

Note that this formula assumes no degeneracy. If there is degeneracy, the formula you give has to be multiplied by the degree of degeneracy to give the expectation value of particle number for energy $E$.

Of course, but it seemed to me that this is not what confused the OP.

 

[PeterDonis]

it seemed to me that this is not what confused the OP.

On the contrary, I think that is exactly what confused the OP; he is asking about a system with degeneracy but appears to believe that the number of particles with energy $E$ can't be more than 1 even if there is degeneracy (multiple states with the same energy $E$). See my post #4, and the portion of post #3 that I responded to there.

 

[Demystifier]

On the contrary, I think that is exactly what confused the OP; he is asking about a system with degeneracy but appears to believe that the number of particles with energy $E$ can't be more than 1 even if there is degeneracy (multiple states with the same energy $E$). See my post #4, and the portion of post #3 that I responded to there.

When I read it again, I see that you are right.