Phy5645/angularmomcommutation/: Difference between revisions

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(New page: '''Question:''' In the angular momentum basis, compute <math>\left \langle {l,m\left |{L_{x}^{2}} \right |l,m} \right \rangle </math> and <math>\left \langle {l,m\left |{L_{x}L_{y}} \righ...)
 
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Again, <math>L_{+}^{2}</math> and <math>L_{-}^{2}</math> won't contribute
Again, <math>L_{+}^{2}</math> and <math>L_{-}^{2}</math> won't contribute


<math>=\frac{1}{4i}\left \langle {lm\left |{L_{-}L_{+}-L_{+}L_{-}} \right |lm} \right \rangle =\frac{1}{4i}\left \lbrace {\left \langle {lm\left |{L_{-}L_{+}} \right |lm} \right \rangle -\left \langle {lm\left |{L_{+}L_{-}} \right |lm} \right \rangle } \right \rbrace </math>
<math>=\frac{1}{4i}\left \langle {l,m\left |{L_{-}L_{+}-L_{+}L_{-}} \right |l,m} \right \rangle =\frac{1}{4i}\left \lbrace {\left \langle {l,m\left |{L_{-}L_{+}} \right |l,m} \right \rangle -\left \langle {l,m\left |{L_{+}L_{-}} \right |l,m} \right \rangle } \right \rbrace </math>


<math>=\frac{\hbar }{4i}\left \lbrace {\sqrt {l(l+1)-m(m+1)} \left \langle {lm\left |{L_{-}} \right |lm+1} \right \rangle -\sqrt {l(l+1)-m(m-1)} \left \langle {lm\left |{L_{+}} \right |lm-1} \right \rangle } \right \rbrace </math>
<math>=\frac{\hbar }{4i}\left \lbrace {\sqrt {l(l+1)-m(m+1)} \left \langle {l,m\left |{L_{-}} \right |l,m+1} \right \rangle -\sqrt {l(l+1)-m(m-1)} \left \langle {l,m\left |{L_{+}} \right |l,m-1} \right \rangle } \right \rbrace </math>


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


<math>=\frac{\hbar ^{2}}{4i}\left ({l(l+1)-m^{2}-m-l(l+1)+m^{2}-m} \right )</math>
<math>=\frac{\hbar ^{2}}{4i}\left ({l(l+1)-m^{2}-m-l(l+1)+m^{2}-m} \right )</math>

Revision as of 00:35, 26 October 2009

Question: In the angular momentum basis, compute 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 } 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 \left \langle {l,m\left |{L_{x}L_{y}} \right |l,m} \right \rangle } .

Solution:

  • 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 L_{x}=\frac{L_{+}+L_{-}}{2}} , we can obtain 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}} ;

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}=\frac{1}{4}(L_{+}+L_{-})^{2}=\frac{1}{4}(L_{+}^{2}-L_{-}^{2}+2L_{+}L_{-}+2L_{-}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 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 L_{-}^{2}} won't contribute because 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_{+}^{2}} \right |l,m} \right \rangle =\left \langle {l,m\left | {l,m+2} \right. } \right \rangle =0} 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 \left \langle {l,m\left |{L_{-}^{2}} \right |l,m} \right \rangle =\left \langle {l,m\left | {l,m-2} \right. } \right \rangle =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 \left \langle {l,m\left |{L_{x}^{2}} \right |l,m} \right \rangle =\frac{1}{4}\left \lbrace {\left \langle {l,m\left |{2L_{+}L_{-}+2L_{-}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{1}{2}\left \lbrace {\left \langle {l,m\left |{L_{+}L_{-}} \right |l,m} \right \rangle +\left \langle {l,m\left |{L_{-}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 \text{=}\frac{\hbar }{2}\sqrt {l(l+1)-m(m-1)} \left \langle {l,m\left |{L_{+}} \right |l,m-1} \right \rangle +\frac{\hbar }{2}\sqrt {l(l+1)-m(m+1)} \left \langle {l,m\left |{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 \text{=}\frac{\hbar ^{2}}{2}\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}}{2}\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}}{2}\left \lbrace {l(l+1)-m(m-1)+l(l+1)-m(m-1)} \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}\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 =\hbar ^{2}(l(l+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 \left \langle {l,m\left |{L_{x}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 L_{x}L_{y}=\frac{L_{+}+L_{-}}{2}\frac{L_{+}-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}(L_{+}+L_{-})(L_{+}-L_{-})=\frac{1}{4i}\left \lbrace {L_{+}^{2}-L_{-}^{2}-L_{+}L_{-}+L_{-}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 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 L_{-}^{2}} won't contribute

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 |{L_{-}L_{+}-L_{+}L_{-}} \right |l,m} \right \rangle =\frac{1}{4i}\left \lbrace {\left \langle {l,m\left |{L_{-}L_{+}} \right |l,m} \right \rangle -\left \langle {l,m\left |{L_{+}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 |{L_{-}} \right |l,m+1} \right \rangle -\sqrt {l(l+1)-m(m-1)} \left \langle {l,m\left |{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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