homework and exercises - Totally antisymetric wavefunction: clarification about terminology

Pauli's Principle says:

"The wavefunction of two identical fermions must be totally antisymmetric".

I know that, for a antisymmetric wavefunction,

$(-1)^L*(-1)^{S+1}*(-1)^{I+1}=-1$

"totally antisymmetric" means this relation or it means that these 3 relations:

$(-1)^L=-1$ and

$(-1)^{S+1}=-1$ and

$(-1)^{I+1}=-1$

must be verified simultaneusly?

Answer

It's the total product. The famous example is the spin of the deuteron. We have evidence that the two-nucleon isospin triplet with $I=1$ is unbound because we do not observe diprotons or dineutrons in nature, so we expect the deuteron to have isospin $I=0$. We know that the deuteron has positive parity, so we require $L$ even; by antisymmetry the deuteron must have spin $S=1$.