How Does the Exclusion Principle Affect Energy Levels in Electron Spin States?

[alle.fabbri]

Hi guys!
When we consider a system composed by two 1/2 spin particle we can label the 4 natural basis vector by the individual spin of each particle, i.e. |++>,|+->,... , or by the eigenvalues of the total spin S and its projection M. In the latter case we have again 4 basis vectors: a singlet state |00> and the triplet states |11> |10> |1-1>.

Now consider the previous particles to be electrons in an infinite square well of size L so the system can be either in the singlet state or in one of the triplet states. If I want to evaluate the energies of the states do I have to consider the exclusion principle?
To be more explicit let's call E the energy of the single particle ground state inside the well, in this way 4E will be the energy of the second, 9E that of the third and so on. My question then becomes, considering for instance |11>, what is the energy? I don't know if I have to take into account the exclusion principle and thus the energy should be 5E, i.e. E from the ground state particle and 4E from the first excited level one since two "up" electrons cannot share the same energy level, or not.
And in a case such |10> or |00>, where it results impossible to assign a level to each particle since the system is in superposition of the individual spin eigenvectors how do I have to proceed?

Thanks for any help...

 

[Meir Achuz]

The Fermi-Dirac statistics of two identical Fermions means that two electrons in the singlet state must be in an even L state, which includes the lowest state in the box.
Two electrons in the triplet state must be in an odd L state, so their lowest state will lie higher.

The |00> state is a pure, not a mixed state, even if the individual spins differ. The |10> state has the same spatial wave function as the |11> and the |1,-1> states.