Phy5645/angularmomcommutation/

(a) 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 \left \langle {l,m\left |{L_{x}^{2}} \right |l,m} \right \rangle }

Since 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 \hat{L}_{x}=\frac{\hat{L}_{+}+\hat{L}_{-}}{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 L_{x}^{2}=\tfrac{1}{4}(\hat{L}_{+}+\hat{L}_{-})^{2}=\tfrac{1}{4}(\hat{L}_{+}^{2}+\hat{L}_{-}^{2}+\hat{L}_{+}\hat{L}_{-}+\hat{L}_{-}\hat{L}_{+}).}

By definition, 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 \hat{L}_{+}^{2}} 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 \hat{L}_{-}^{2}} won't contribute because they reduce to an inner product of the form, 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 \left \langle {l,m\left | {l,m\pm 2} \right. } \right \rangle =0.} Therefore, the only contribution is from the last two terms:

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 \left \langle {l,m\left |{\hat{L}_{x}^{2}} \right |l,m} \right \rangle =\tfrac{1}{4}\left \lbrace {\left \langle {l,m\left |{\hat{L}_{+}\hat{L}_{-}+\hat{L}_{-}\hat{L}_{+}} \right |l,m} \right \rangle } \right \rbrace }

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 =\tfrac{1}{4}\left \lbrace {\left \langle {l,m\left |{\hat{L}_{+}\hat{L}_{-}} \right |l,m} \right \rangle +\left \langle {l,m\left |{\hat{L}_{-}\hat{L}_{+}} \right |l,m} \right \rangle } \right \rbrace }

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 =\frac{\hbar }{4}\sqrt {l(l+1)-m(m-1)} \left \langle {l,m\left |{\hat{L}_{+}} \right |l,m-1} \right \rangle +\frac{\hbar }{4}\sqrt {l(l+1)-m(m+1)} \left \langle {l,m\left |{\hat{L}_{-}} \right |l,m+1} \right \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 =\frac{\hbar ^{2}}{4}\sqrt {l(l+1)-m(m-1)} \sqrt {l(l+1)-(m-1)m} \left \langle {l,m} \right | \left. {} {l,m} \right \rangle +\frac{\hbar ^{2}}{4}\sqrt {l(l+1)-m(m+1)} \sqrt {l(l+1)-(m+1)m} \left \langle {l,m} \right | \left. {} {l,m} \right \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 =\frac{\hbar ^{2}}{4}\left \lbrace {l(l+1)-m(m-1)+l(l+1)-m(m+1)} \right \rbrace=\frac{\hbar ^{2}}{4}\left \lbrace {2l(l+1)-2m^{2}} \right \rbrace }

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 =\frac{\hbar ^{2}}{2}[l(l+1)-m^{2}]}

(b) 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 \left \langle {l,m\left |{\hat{L}_{x}\hat{L}_{y}} \right |l,m} \right \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 \hat{L}_{x}\hat{L}_{y}=\frac{\hat{L}_{+}+\hat{L}_{-}}{2}\frac{\hat{L}_{+}-\hat{L}_{-}}{2i}}

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 =\frac{1}{4i}(\hat{L}_{+}+\hat{L}_{-})(\hat{L}_{+}-\hat{L}_{-})=\frac{1}{4i}\left \lbrace {\hat{L}_{+}^{2}-\hat{L}_{-}^{2}-\hat{L}_{+}\hat{L}_{-}+\hat{L}_{-}\hat{L}_{+}} \right \rbrace }

Again, 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 \hat{L}_{+}^{2}} 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 \hat{L}_{-}^{2}} won't contribute. We now evaluate the remaining terms:

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 =\frac{1}{4i}\left \langle {l,m\left |{\hat{L}_{-}\hat{L}_{+}-\hat{L}_{+}\hat{L}_{-}} \right |l,m} \right \rangle =\frac{1}{4i}\left \lbrace {\left \langle {l,m\left |{\hat{L}_{-}\hat{L}_{+}} \right |l,m} \right \rangle -\left \langle {l,m\left |{\hat{L}_{+}\hat{L}_{-}} \right |l,m} \right \rangle } \right \rbrace }

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 =\frac{\hbar }{4i}\left \lbrace {\sqrt {l(l+1)-m(m+1)} \left \langle {l,m\left |{\hat{L}_{-}} \right |l,m+1} \right \rangle -\sqrt {l(l+1)-m(m-1)} \left \langle {l,m\left |{\hat{L}_{+}} \right |l,m-1} \right \rangle } \right \rbrace }

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 =\frac{\hbar ^{2}}{4i}\left \lbrace {\sqrt {l(l+1)-m(m+1)} \sqrt {l(l+1)-(m+1)m} \left \langle {l,m} \right | \left. {} {l,m} \right \rangle -\sqrt {l(l+1)-m(m-1)} \sqrt {l(l+1)-(m-1)m} \left \langle {l,m} \right | \left. {} {l,m} \right \rangle } \right \rbrace }

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 =\frac{\hbar ^{2}}{4i}\left ({l(l+1)-m^{2}-m-l(l+1)+m^{2}-m} \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 =\frac{\hbar ^{2}}{4i}(-2m)=\frac{i\hbar ^{2}m}{2}}

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