Identical particles present us with another symmetry in nature.
Electrons, for example, are indistinguishable from each other so we must have
symmetry of the Hamiltonian under interchange
of any pair of electrons.
Lets call the operator that interchanges electron-1 and electron-2
So we can make our energy eigenstates also eigenstates of
Its easy to see (by operating on an eigenstate twice with
that the possible eigenvalues are
It is a law of physics that spin
particles called fermions
(like electrons) always are antisymmetric under interchange,
while particles with integer spin called bosons (like photons)
always are symmetric under interchange.
Antisymmetry under interchange leads to the Pauli exclusion principle that no two
electrons (for example) can be in the same state.