Phy5646/CG coeff example1: Difference between revisions
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== Find the CG coefficients == | == Find the CG coefficients == | ||
<math> 1\rangle : \dfrac{1}{2}\otimes 1 = \dfrac{3}{2}\oplus \dfrac{1}{2} </math> | <math> 1\rangle : \dfrac{1}{2}\otimes 1 = \dfrac{3}{2}\oplus \dfrac{1}{2} </math> <math>\left [ Shankar excercise: 15.2.2 \right ]</math> | ||
'''Answer''' | '''Answer''' | ||
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The state <math>|\dfrac{3}{2},\dfrac{3}{2} \rangle</math> is given by, | The state <math>|\dfrac{3}{2},\dfrac{3}{2} \rangle</math> is given by, | ||
<math>|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{2}\rangle</math> = <math>|\dfrac{1}{2}, 1, \dfrac{1}{2},1\rangle</math> | |||
Corresponding CG coefficient, <math>\langle\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}</math> <math>|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{2}\rangle</math> = 1 | Corresponding CG coefficient, <math>\langle\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}</math> <math>|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{2}\rangle</math> = 1 | ||
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<math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle</math> + <math>\sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle</math> | <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle</math> + <math>\sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle</math> | ||
CG coefficients: | |||
<math>\langle \dfrac{1}{2}, 1, -\dfrac{1}{2},1</math> <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{1}{3}}</math> | <math>\langle \dfrac{1}{2}, 1, -\dfrac{1}{2},1</math> <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{1}{3}}</math> | ||
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<math>\langle \dfrac{1}{2}, 1, \dfrac{1}{2},0</math> <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{2}{3}}</math> | <math>\langle \dfrac{1}{2}, 1, \dfrac{1}{2},0</math> <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> = <math>\sqrt{\frac{2}{3}}</math> | ||
Similarly | Similarly by repeated application of <math>J_{-}</math> and <math>J_{1-}</math>,<math>J_{2-}</math> on <math> |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> we get, | ||
<math> J_{-}|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math>= <math>2\hbar |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math> | <math> J_{-}|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math>= <math>2\hbar |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math> | ||
and <math> (J_{1-}+ J_{-})|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math>= <math>(J_{1-}+ J_{-}) \left [\sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle + \sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle \right ]</math> | and <math> (J_{1-}+ J_{-})|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math>= <math>(J_{1-}+ J_{-}) \left [\sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle + \sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle \right ]</math> | ||
<math> (J_{1-}+ J_{2-})|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math>= <math>2\sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},0\rangle + 2\sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},-1\rangle </math> | |||
<math> |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math>= <math>\sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},0\rangle + \sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},-1\rangle </math> | |||
CG coefficients: | |||
<math>\langle \dfrac{1}{2}, 1, -\dfrac{1}{2},0 |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math>= <math>\sqrt{\frac{2}{3}} </math> | |||
<math>\langle \dfrac{1}{2}, 1, \dfrac{1}{2},-1 |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math>= <math>\sqrt{\frac{1}{3}} </math> | |||
Again by repeated application of <math>J_{-}</math> and <math>J_{1-}</math>,<math>J_{2-}</math> | |||
<math> J_{-}|\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{1}{2}\rangle</math>= <math>\hbar\sqrt{3} |\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{3}{2}\rangle </math> | |||
<math>(J_{1-}+ J_{2-})\left [\sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{2},0\rangle + \sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},-1\rangle \right ] </math>= <math>\hbar\sqrt{3} |\dfrac{1}{2}, 1, -\dfrac{1}{2},-1\rangle </math> | |||
CG coefficients: | |||
<math>\langle\dfrac{1}{2}, 1, -\dfrac{1}{2},-1</math> <math>|\dfrac{1}{2}, 1, \dfrac{3}{2},-\dfrac{3}{2}\rangle </math> = 1 | |||
'''Eigenvectors <math>|j m \rangle</math> associated with <math>j= \dfrac{1}{2}</math>''': | |||
<math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math>= <math> a|\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle </math> <math> +b|\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle </math> | |||
where, <math> a</math> <math>= \langle\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}|\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle </math> | |||
<math> b</math> <math>= \langle\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}|\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle </math> | |||
Therefore ,<math>\langle\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}</math><math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math> = <math> a^{2} + b^{2} = 1</math> | |||
Since, <math>|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle</math> and <math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math> are orthogonal to each other, <math>\langle\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}</math><math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math> = <math> a\sqrt{\frac{2}{3}} + b\sqrt{\frac{1}{3}} = 0</math> | |||
<math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math>= <math> -\sqrt{\frac{1}{3}}|\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle </math> <math> +\sqrt{\frac{2}{3}}|\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle </math> | |||
CG coefficients: | |||
<math>\langle\dfrac{1}{2}, 1, \dfrac{1}{2},0</math><math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math>= <math> -\sqrt{\frac{1}{3}}</math> | |||
<math>\langle\dfrac{1}{2}, 1, -\dfrac{1}{2},1</math><math>|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math>= <math> \sqrt{\frac{2}{3}}</math> | |||
Again by repeated application of <math>J_{-}</math> and <math>J_{1-}</math>,<math>J_{2-}</math> | |||
<math>J_{-}|\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{1}{2}\rangle</math>= <math> \hbar|\dfrac{1}{2}, 1, \dfrac{1}{2},-\dfrac{1}{2}\rangle </math> | |||
<math> (J_{1-}+ J_{2-})\left [-\sqrt{\frac{1}{3}}|\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle +\sqrt{\frac{2}{3}}|\dfrac{1}{2}, 1, -\dfrac{1}{2},1\rangle \right ] </math> = <math>\hbar \left [\sqrt{\frac{1}{3}}|\dfrac{1}{2}, 1, -\dfrac{1}{2},0\rangle - \sqrt{\frac{2}{3}}|\dfrac{1}{2}, 1, \dfrac{1}{2},-1\rangle \right ]</math> | |||
<math> |\dfrac{1}{2}, 1, \dfrac{1}{2},-\dfrac{1}{2}\rangle </math> = <math> \left [\sqrt{\frac{1}{3}}|\dfrac{1}{2}, 1, -\dfrac{1}{2},0\rangle - \sqrt{\frac{2}{3}}|\dfrac{1}{2}, 1, \dfrac{1}{2},-1\rangle \right ]</math> | |||
CG coefficients: | |||
<math>\langle\dfrac{1}{2}, 1, -\dfrac{1}{2},0 |\dfrac{1}{2}, 1, \dfrac{1}{2},-\dfrac{1}{2}\rangle </math> = <math> \sqrt{\frac{1}{3}}</math> | |||
<math>\langle\dfrac{1}{2}, 1, \dfrac{1}{2}, -1|\dfrac{1}{2}, 1, \dfrac{1}{2},-\dfrac{1}{2}\rangle </math> = <math> -\sqrt{\frac{1}{3}}</math> | |||
This completes the analysis :o) | |||
Latest revision as of 22:20, 30 April 2010
Find the CG coefficients
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 1\rangle : \dfrac{1}{2}\otimes 1 = \dfrac{3}{2}\oplus \dfrac{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 \left [ Shankar excercise: 15.2.2 \right ]}
Answer
The addition 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 j_1=s=\frac{1}{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 j_2=l=1} is encountered, for example, in the p-state of an electron. This state is characterised by orbital quantum number 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 j_1=s=\frac{1}{2}} and spin quantum number 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 j_2=l=1} . Obviously the possible values of magnetic quantum number for 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 j_1=s=\frac{1}{2}} 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 m_s = \frac{1}{2},-\frac{1}{2}} and those for 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 j_2=l=1} 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 m_l = 1,0,-1} . The allowed values of the total angular momentum are between 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 |j_1 - j_2| \le j \le j_1 + j_2 } hence 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 j = \frac{3}{2},\frac{1}{2}} . To calculate the relevant Clebsch–Gordan coefficients, we have to express the basis vectors 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 |j m j_1 j_2\rangle} 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 |\dfrac{1}{2}, 1, m_1 m_2\rangle}
Eigenvectors 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 |j m \rangle} associated 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 j= \dfrac{3}{2}} :
The 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 |\dfrac{3}{2},\dfrac{3}{2} \rangle} is given by,
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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{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 |\dfrac{1}{2}, 1, \dfrac{1}{2},1\rangle}
Corresponding CG coefficient, 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\dfrac{1}{2}, 1, \dfrac{1}{2},\dfrac{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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{2}\rangle} = 1
Now 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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle} can be found by
Applying 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 J_{-} } 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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{2}\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 (J_{1-} +J_{2-})} 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 |\dfrac{1}{2}, 1, \dfrac{1}{2},1\rangle} and the equating the two results,
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 J_{-}|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{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 (J_{1-} +J_{2-})|\dfrac{1}{2}, 1, \dfrac{1}{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 J_{-} |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{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 \sqrt{\frac{3}{2}\left ( \frac{3}{2}+1 \right )-\frac{3}{2}\left ( \frac{3}{2}-1 \right )}|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle}
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 J_{-} |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{3}{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 \sqrt{3}|\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{2}\rangle}
Now 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 (J_{1-} +J_{2-})|\dfrac{1}{2}, 1, \dfrac{1}{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 \sqrt{\frac{1}{2}\left ( \frac{1}{2}+1 \right )-\frac{1}{2}\left ( \frac{1}{2}-1 \right )} |\dfrac{1}{2}, 1, -\dfrac{1}{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 \sqrt{1\left ( 1+1 \right )-1\left ( 1-1 \right )} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\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 (J_{1-} +J_{2-})|\dfrac{1}{2}, 1, \dfrac{1}{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 |\dfrac{1}{2}, 1, -\dfrac{1}{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 \sqrt{2} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{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 \sqrt{\frac{1}{3}} |\dfrac{1}{2}, 1, -\dfrac{1}{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 \sqrt{\frac{2}{3}} |\dfrac{1}{2}, 1, \dfrac{1}{2},0\rangle}
CG coefficients:
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 \dfrac{1}{2}, 1, -\dfrac{1}{2},1} 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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{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 \sqrt{\frac{1}{3}}}
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 \dfrac{1}{2}, 1, \dfrac{1}{2},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 |\dfrac{1}{2}, 1, \dfrac{3}{2},\dfrac{1}{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 \sqrt{\frac{2}{3}}}
Similarly by repeated application 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 J_{-}} 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 J_{1-}} ,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 J_{2-}} on we get,
=
and =
![{\displaystyle (J_{1-}+J_{-})\left[{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},1\rangle +{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},0\rangle \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9687a71f3e80665bb3d0bc2fbad6f9b2097dd3d)
=
=
CG coefficients:
=
=
Again by repeated application of and ,
=
=
![{\displaystyle (J_{1-}+J_{2-})\left[{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle +{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3131c1459a0e6df9bbe3e38d2678468d22fe8963)
CG coefficients:
= 1
Eigenvectors associated with :
=
where,
Therefore , =
Since, and are orthogonal to each other, =
=
CG coefficients:
=
=
Again by repeated application of and ,
=
=
![{\displaystyle (J_{1-}+J_{2-})\left[-{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},0\rangle +{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},1\rangle \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6792fd6d8439511bf0c2ce2c75a6d7f907590ae)
![{\displaystyle \hbar \left[{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle -{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/037598c5a38d6460b6b2b274689c23644dc24644)
=
![{\displaystyle \left[{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle -{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db7ef74aad78f98333de4c52f2f08859336d68f5)
CG coefficients:
=
=
This completes the analysis :o)