Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section C / C11 / c

Addition of two spin-one-half angular momenta gives total spin $S=0$ or $S=1$. The normalized singlet state and triplet states areThey have $S^2$ eigenvalues $0$ and $2{\hslash }^2$, and $S_z=M\hslash$. The singlet state is antisymmetric under a particle exchange operator, whereas all three triplets have symmetric spin-exchange parity. For two identical electrons the full state must be antisymmetric: a singlet state pairs with a spatial state with symmetric exchange parity, a triplet state with an antisymmetric spatial state. Thus two electrons in the same atomic orbital must form a singlet state.

The identity ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=(S^2-S_1^2-S_2^2)/2$ gives $-3{\hslash }^2/4$ in the singlet state and $+{\hslash }^2/4$ in the triplet state. Hence an exchange interaction distinguishes their energies, with ordering determined by its coupling sign. The singlet state has perfect opposite-spin correlations along any common measurement axis and is entangled. The $M=0$ triplet state is also entangled, while the $M=\pm 1$ triplet states are product states along the chosen axis.