Phy5646/AddAngularMomentumProb

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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} , 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}{4}-m_1(m_1+1)\right)\left(\frac{3}{4}-m_2(m_2-1)\right)}|m_1+1;m_2-1\rangle+\sqrt{\left(\frac{3}{4}-m_1(m_1-1)\right)\left(\frac{3}{4}-m_2(m_2+1)\right)}|m_1-1;m_2+1\rangle\right)}

3.) Now acting on the left with 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 \ \langle m_1'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 \ \langle m_1'm_2'|S^2|m_1m_2\rangle = \hbar^2 \left(\left(\frac{3}{2}+2m_1m_2\right)\delta_{m_1'm_1}\delta_{m_2'm_2}+\sqrt{\left(\frac{3}{4}-m_1(m_1+1)\right)\left(\frac{3}{4}-m_2(m_2-1)\right)}\delta_{m_1'm_1+1}\delta_{m_2'm_2-1}+\sqrt{\left(\frac{3}{4}-m_1(m_1-1)\right)\left(\frac{3}{4}-m_2(m_2+1)\right)}\delta_{m_1'm_1-1}\delta_{m_2'm_2+1}\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 \Rightarrow \langle m_1'm_2'|S^2|m_1m_2\rangle = \hbar^2 \left(\left(\frac{3}{2}+2m_1m_2\right)\delta_{m_1'm_1}\delta_{m_2'm_2}+\sqrt{\left(\frac{3}{4}-m_1m_1'\right)\left(\frac{3}{4}-m_2m_2'\right)}\left(\delta_{m_1'm_1+1}\delta_{m_2'm_2-1}+\delta_{m_1'm_1-1}\delta_{m_2'm_2+1}\right)\right) }

3.) Now plugging 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, m_2, 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 \ \langle 1/2;1/2|S^2|1/2;1/2\rangle = \hbar^2\left(\left(\frac{3}{2}+2\cdot\frac{1}{2}\cdot\frac{1}{2}\right)+0\right) = 2\hbar^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 \ \langle -1/2;-1/2|S^2|-1/2;-1/2\rangle = \hbar^2\left(\left(\frac{3}{2}+2\cdot\frac{-1}{2}\cdot\frac{-1}{2}\right)+0\right) = 2\hbar^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 \langle 1/2;-1/2|S^2|1/2;-1/2\rangle = \left(\left(\frac{3}{2}+2\cdot\frac{1}{2}\cdot\frac{-1}{2}\right)+0\right) = \hbar^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 \langle -1/2;1/2|S^2|-1/2;1/2\rangle = \left(\left(\frac{3}{2}+2\cdot\frac{-1}{2}\cdot\frac{1}{2}\right)+0\right) = \hbar^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 \langle -1/2;1/2|S^2|1/2;1/2\rangle = \langle 1/2;1/2|S^2|-1/2;1/2\rangle = \left(0+0\right) = 0}

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 \langle 1/2;-1/2|S^2|1/2;1/2\rangle = \langle 1/2;1/2|S^2|1/2;-1/2\rangle = \left(0+0\right) = 0}

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 \langle -1/2;-1/2|S^2|1/2;1/2\rangle = \langle 1/2;1/2|S^2|-1/2;-1/2\rangle = \left(0+0\right) = 0}

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 \langle -1/2;1/2|S^2|1/2;-1/2\rangle = \langle 1/2;-1/2|S^2|-1/2;1/2\rangle = \left(0+\sqrt{\left(\frac{3}{4}-\frac{1}{2}\cdot\frac{-1}{2}\right)\left(\frac{3}{4}-\frac{-1}{2}\cdot\frac{1}{2}\right)}\left(0+1\right)\right) = \hbar^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 \langle -1/2;-1/2|S^2|-1/2;1/2\rangle = \langle -1/2;1/2|S^2|-1/2;-1/2\rangle = \left(0+0\right) = 0}

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 \langle -1/2;-1/2|S^2|1/2;-1/2\rangle = \langle 1/2;-1/2|S^2|-1/2;-1/2\rangle = \left(0+0\right) = 0}


All that must be done now is arranging the matrix elements in matrix form. The ++ state corresponds to the left (top) while the -- state is on the right (bottom):


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

We can clearly see that all of the direct product states do not diagonalize 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 } . A linear combination of the two problem states, 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 \ |+-\rangle } 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 \ |-+\rangle } , should solve the problem however:


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(\alpha|+-\rangle + \beta|-+\rangle) = \lambda(\alpha|+-\rangle + \beta|-+\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_1^2 + S_2^2 + 2S_{1z}S_{2z}+S_{1+}S_{2-}+S_{1-}S_{2+})(\alpha|+-\rangle + \beta|-+\rangle) = \lambda(\alpha|+-\rangle + \beta|-+\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 \hbar^2\alpha\left(\left(\frac{3}{4} + \frac{3}{4} + 2\left(\frac{1}{2}\cdot\frac{-1}{2}\right)\right)|+-\rangle + 0 + \sqrt{\left(\frac{3}{4}+\frac{1}{4}\right)\left(\frac{3}{4}+\frac{1}{4}\right)}|-+\rangle\right) + \hbar^2\beta\left(\left(\frac{3}{4} + \frac{3}{4} + 2\left(\frac{-1}{2}\cdot\frac{1}{2}\right)\right)|-+\rangle + \sqrt{\left(\frac{3}{4}+\frac{1}{4}\right)\left(\frac{3}{4}+\frac{1}{4}\right)}|+-\rangle + 0\right) = \lambda(\alpha|+-\rangle + \beta|-+\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\hbar^2\alpha\left(|+-\rangle + |-+\rangle\right) + \hbar^2\beta\left(|-+\rangle + |+-\rangle)\right) = \lambda(\alpha|+-\rangle + \beta|-+\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\hbar^2\left(\alpha+\beta\right)\cdot\left(|+-\rangle + |-+\rangle\right) = \lambda(\alpha|+-\rangle + \beta|-+\rangle) }

The only two possibilities are 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 \alpha = \plusmn\beta} . In the 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 \ \alpha = +\beta} case we get 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 \lambda = 2\hbar^2} which corresponds to 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} . In the 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 \ \alpha = -\beta} case we get 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 \ \lambda = 0} which corresponds to 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 = 0} . The normalization condition requires 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 \ \alpha = \beta = (2)^{-1/2}} or 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 \ \alpha = (2)^{-1/2}, \beta = -(2)^{-1/2}} .

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