Phy5646/AddAngularMomentumProb

Based on exercise 15.1.1. from Principles of Quantum Mechanics, 2nd ed. by Shankar:

Express $\ S^{2}$ as a matrix for two spin-1/2 particles in the direct product basis.

1.) First express $\ S^{2}$ in terms of $\ S_{1}^{2}$, $\ S_{2}^{2}$, $\ S_{1z}$, $\ S_{2z}$, $\ S_{1\pm }$ and $\ S_{2\pm }$: $\ S^{2}=({\vec {S_{1}}}+{\vec {S_{2}}})^{2}=S_{1}^{2}+S_{2}^{2}+2{\vec {S_{1}}}\cdot {\vec {S_{2}}}=S_{1}^{2}+S_{2}^{2}+2(S_{1x}S_{2x}+S_{1y}S_{2y}+S_{1z}S_{2z})=S_{1}^{2}+S_{2}^{2}+2S_{1z}S_{2z}+S_{1+}S_{2-}+S_{1-}S_{2+}$.

2.) Act with this on a 2-electron state with arbitrary $\ S_{z}$ quantum numbers:

${\hbar }^{2}{\begin{pmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&2\\\end{pmatrix}}$