Configurations for Non-Interacting Particles with Total Energy of 2E

[tomwilliam]

Homework Statement

List all possible configurations for distinguishable particles, identical bosons and identical fermions where the system has a total energy of 2E and contains two non-interacting particles. The energy-level structure is 0, 1ER, 1EL and 2E (where 1ER and 1EL are degenerate).

Homework Equations

The Attempt at a Solution

I know the situation for distinguishable particles.
I think I know that bosons can be in the same quantum state, so that includes the possibility of the two particles both being in 1ER and then in 1ER (I make it four configurations in total).
What I don't know is whether the Pauli exclusion principle makes it impossible for the two fermions to be at the same energy level, if there are two degenerate levels with the same energy. Is it possible for them to be at a) 2E and 0, and b) 1ER and 1EL...or does the exclusion principle cancel that possibility?
Thanks

 

[vela]

The Pauli exclusion principle says that two fermions can't be in the same state at the same time. Since 1ER and 1EL are different states, the Pauli exclusion principle doesn't apply.

 

[tomwilliam]

Thanks
So two fermions in the same system can have the same energy...providing they have a different quantum state?

 

[vela]

Yup, like two electrons in the n=1 state of the hydrogen atom.

 

[paranormal]

for your question. I can confirm that your understanding of the configurations for distinguishable particles and bosons is correct. However, for fermions, the Pauli exclusion principle does indeed prevent them from occupying the same energy level. This means that for two non-interacting fermions with a total energy of 2E, the only possible configurations are 1ER and 1EL (with one particle in each energy level) or 2E and 0 (with one particle in the 2E level and the other in the 0 level). This is because the exclusion principle states that no two fermions can have the same set of quantum numbers, including energy level. Therefore, the possibility of both fermions being in the 1ER level is excluded. I hope this helps clarify the situation for you. Keep up the good work in your studies!