No two identical fermions can have the same quantum state at once?

Question

This is the Pauli Exclusion Principle, but I have a question about it... It states that no two identical fermions can have the same quantum state, but what about different fermions having the same quantum state?

Answer 1

A muon and a positron are different species so the wavefunction need not be symmetric or antisymmetric under interchange of the positions of the two different particles.

That's good since the momentum operator takes the derivative in that particles direction and then scales by that particles mass, so it would be weird if they swapped.

So you can have an infinite square well, have both in the ground state and have them both be spin up. Of course that assumes they don't interact, but an interaction won't make their wavefunction antisymmetric it will just give them a potential between the two.

Answer 2

Pauli principle says that fermions have antisymmetric wave function, from which it follows that they cannot be in the same state: $$\phi(x_1,x_2)=-\phi(x_2,x_1)\Rightarrow \phi(x,x)=-\phi(x,x)=0.$$

This follows more generally from the fact that for identical particles the probability amplitude should be invariant under permutations: $$ |\phi(x_1,x_2)|^2=|\phi(x_2,x_1)|^2\Rightarrow \phi(x_1,x_2)=\pm\phi(x_2,x_1).$$

Note that the interpretation in terms of the impossibility to be in the same state is more meaningful in case of non-interacting fermions, where the joint two-particle wave function can be decomposed into one-particle wave functions: $$ \phi(x_1,x_2)=\varphi(x_1)\psi(x_2)-\psi(x_1)\varphi(x_2) $$ In interacting case the quantum numbers (i.e., state) really refer to the state of the two-particle system, although one could still perhaps interpret the wave function in terms of one-particle operators, with eigenstates labeled by quantum numbers $x_1,x_2$.

Note also that $x_1,x_2$ are not only position, but all the quantum numbers characterizing single particle - a classical case here is constructing the hydrigen molecule or Helium atom eigenstates out of one-particle eigenstates, where either orbital or spin components of the wave function has to be anti-symmetric, which results in singlet/triplet distinction.

Finally, nothing prevents the fermions from being extended in space, the Pauli principle only means that the wave function has nodes, when two coordinates are equal.