Identical Particles

Identical particles present us with another symmetry in nature. Electrons, for example, are indistinguishable from each other so we must have a symmetry of the Hamiltonian under interchange of any pair of electrons. Lets call the operator that interchanges electron-1 and electron-2 $P_{12}$.

$$\bgroup\color{black}[H,P_{12}]=0\egroup$$

So we can make our energy eigenstates also eigenstates of $P_{12}$. Its easy to see (by operating on an eigenstate twice with $P_{12}$), that the possible eigenvalues are $\pm 1$. It is a law of physics that spin ${1\over 2}$ 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.