Phy5646/AddAngularMomentumProb

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

Express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ S^2 } as a matrix for two spin-1/2 particles in the direct product basis.

1.) First express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ S^2} in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 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.) Then act with this on direct product state $\ |m_{1}m_{2}\rangle$: $\ S^{2}|m_{1}m_{2}\rangle =(S_{1}^{2}+S_{2}^{2}+2S_{1z}S_{2z}+S_{1+}S_{2-}+S_{1-}S_{2+})|m_{1}m_{2}\rangle$ $\ =S_{1}^{2}|m_{1}m_{2}\rangle +S_{2}^{2}|m_{1}m_{2}\rangle +2S_{1z}S_{2z}|m_{1}m_{2}\rangle +S_{1+}S_{2-}|m_{1}m_{2}\rangle +S_{1-}S_{2+}|m_{1}m_{2}\rangle$

$\ =\hbar ^{2}{\frac {1}{2}}\left({\frac {1}{2}}+1\right)|m_{1}m_{2}\rangle +\hbar ^{2}{\frac {1}{2}}\left({\frac {1}{2}}+1\right)|m_{1}m_{2}\rangle +2\hbar ^{2}m_{1}m_{2}|m_{1}m_{2}\rangle +\hbar ^{2}{\sqrt {{\frac {1}{2}}\left({\frac {1}{2}}+1\right)-m_{1}\left(m_{1}+1\right)}}{\sqrt {{\frac {1}{2}}\left({\frac {1}{2}}+1\right)-m_{2}\left(m_{1}-1\right)}}|m_{1}+1;m_{2}-1\rangle$ $\ +\hbar ^{2}{\sqrt {{\frac {1}{2}}\left({\frac {1}{2}}+1\right)-m_{1}\left(m_{1}-1\right)}}{\sqrt {{\frac {1}{2}}\left({\frac {1}{2}}+1\right)-m_{2}\left(m_{1}+1\right)}}|m_{1}-1;m_{2}+1\rangle$ $\ \Rightarrow S^{2}|m_{1}m_{2}\rangle =\hbar ^{2}\left(\left({\frac {3}{2}}+2m_{1}m_{2}\right)|m_{1}m_{2}\rangle +{\sqrt {\left({\frac {3}{2}}-m_{1}(m_{1}+1)\right)\left({\frac {3}{2}}-m_{2}(m_{2}-1)\right)}}|m_{1}+1;m_{2}-1\rangle +{\sqrt {\left({\frac {3}{2}}-m_{1}(m_{1}-1)\right)\left({\frac {3}{2}}-m_{2}(m_{2}+1)\right)}}|m_{1}-1;m_{2}+1\rangle \right)$

3.) Now plug in appropriate values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ m_1 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ m_2 } :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\hbar}^2\begin{pmatrix} 2 & 0 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}}