Phy5646/CG coeff example1

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} }

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}

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 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} we get,

$J_{-}|{\dfrac {1}{2}},1,{\dfrac {3}{2}},{\dfrac {1}{2}}\rangle$= $2\hbar |{\dfrac {1}{2}},1,{\dfrac {3}{2}},-{\dfrac {1}{2}}\rangle$

and $(J_{1-}+J_{-})|{\dfrac {1}{2}},1,{\dfrac {3}{2}},{\dfrac {1}{2}}\rangle$= $(J_{1-}+J_{-})\left[{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},1\rangle +{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},0\rangle \right]$

$(J_{1-}+J_{2-})|{\dfrac {1}{2}},1,{\dfrac {3}{2}},{\dfrac {1}{2}}\rangle$= $2{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle +2{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle$

$|{\dfrac {1}{2}},1,{\dfrac {3}{2}},-{\dfrac {1}{2}}\rangle$= ${\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle +{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle$

Again by repeated application of $J_{-}$ and $J_{1-}$,$J_{2-}$

$J_{-}|{\dfrac {1}{2}},1,{\dfrac {3}{2}},-{\dfrac {1}{2}}\rangle$= $\hbar {\sqrt {3}}|{\dfrac {1}{2}},1,{\dfrac {3}{2}},-{\dfrac {3}{2}}\rangle$

$(J_{1-}+J_{2-})\left[{\sqrt {\frac {2}{3}}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},0\rangle +{\sqrt {\frac {1}{3}}}|{\dfrac {1}{2}},1,{\dfrac {1}{2}},-1\rangle \right]$= $\hbar {\sqrt {3}}|{\dfrac {1}{2}},1,-{\dfrac {1}{2}},-1\rangle$

$\langle {\dfrac {1}{2}},1,-{\dfrac {1}{2}},-1$ $|{\dfrac {1}{2}},1,{\dfrac {3}{2}},-{\dfrac {3}{2}}\rangle$ = 1