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 $\ S^{2}$ in terms of $\ S_{1}^{2}$, 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 \ S_2^2} , 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 \ S_{1z}} , 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_{2z}} , 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\plusmn}} 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 \ S_{2\plusmn}} : 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 = (\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 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_1m_2\rangle } : 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|m_1m_2\rangle = (S_1^2 + S_2^2 + 2S_{1z}S_{2z}+S_{1+}S_{2-}+S_{1-}S_{2+})|m_1m_2\rangle } 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|m_1m_2\rangle + S_2^2|m_1m_2\rangle + 2S_{1z}S_{2z}|m_1m_2\rangle+S_{1+}S_{2-}|m_1m_2\rangle+S_{1-}S_{2+}|m_1m_2\rangle }

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 \frac{1}{2}\left(\frac{1}{2}+1 \right)|m_1m_2\rangle + \hbar^2\frac{1}{2}\left(\frac{1}{2}+1\right)|m_1m_2\rangle + 2\hbar^2m_1m_2|m_1m_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} 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\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 } 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 \ \Rightarrow S^2|m_1m_2\rangle = \hbar^2 \left(\left(\frac{3}{2}+2m_1m_2\right)|m_1m_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 \ S^2 |1/2;1/2\rangle = \hbar^2\left(\left(\frac{3}{2}+2\cdot\frac{1}{4}\right)|1/2;1/2\rangle\right) }

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}}