# Pauli Exclusion Principle-question

1. Jun 1, 2007

Pauli Exclusion Principle--question

According to the Pauli Exclusion Principle, two identical FERMIONS cannot coexist in the same energy level within a nucleus.

My question is, could two identical fermions coexist if they had opposite isospin, that is, does isospin make the two otherwise identicle fermions "not-identicle" such that the Pauli EP does not apply? Thanks in advance for any help, I would appreciate mathematical details in any answer.

2. Jun 1, 2007

### mathman

Electrons in atoms come in pairs (except for possibly one left over) with opposite spin, but otherwise identical properties.

3. Jun 2, 2007

### Meir Achuz

It is wrong to say "cannot coexist in the same energy level ".
If the energy level is degerate, then a number of fermions equal to the degeneracy coulld fill the same enrgy level.
The deuteron with n of I3=-1/2 and p of I3=+1/2 is an example of
two nucleons with opposite Ispins n the same space-spin state.

4. Jun 4, 2007

Yes, thank you, of course the P and N "coexist" with opposite isospin as the deuteron--but they are not a priori "identical". So this example you give will not answer the OP question. My question is ...can two identical fermions coexist (use whatever concept you like for energy level such as "space-spin state") if the ONLY difference between them is ISOSPIN... ?

5. Jun 4, 2007

Thank you, but the "spin" of the electron is not the same as "isospin" ---see below from Wiki-- so I do not see how your response in any way helps with the OP question--but thank you for your time:

Heisenberg's contribution was to note that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of spin, from whence the name "isospin" derives. To be precise, the isospin symmetry is given by the invariance of the Hamiltonian of the strong interactions under the action of the Lie group SU(2). The neutron and the proton are assigned to the doublet (the spin-1/2 or fundamental representation) of SU(2). The pions are assigned to the triplet (the spin-1 or adjoint representation) of SU(2).

Just as is the case for regular spin, isospin is described by two numbers, I, the total isospin, and I3, the component of the spin vector in a given direction. The proton and neutron both have I=1/2, as they belong to the doublet. The proton has I3=+1/2 or 'isospin-up' and the neutron has I3=−1/2 or 'isospin-down'. The pions, belonging to the triplet, have I=1, and π+, π0 and π− have, respectively, I3=+1, 0, −1.

6. Jun 4, 2007

### damgo

Depends. Yes, they could, if you're talking about quarks. If you're talking about composite fermions that's another issue.

The key point here is that isospin is not a fundamental quantum number. Rather, it is an approximate symmetry relating two distinct types of fundamental fermions: up and down quarks. Since up and down quarks correspond to different fields, the Pauli exclusion principle does not apply to them together, and they could be in the same state.

7. Jun 4, 2007

### malawi_glenn

You can also look at the motivation for the Shell model of the nucleus, and you will se that protons and neutrons are treated as nonidentical fermions a priori. You will se that for each energy level, 4 nucleons can be contained: 2 protons and 2 neutrons. Were the protons differs by their z-component of intrinsic spin, and also for the neutrons. I think it is quite trivial to see that protons and neutrons are nonidentical particles=)