Phy5646/CG coeff example1: Difference between revisions
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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> |\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> | <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> | Again by repeated application of <math>J_{-}</math> and <math>J_{1-}</math>,<math>J_{2-}</math> | ||
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<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> | <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 | <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>''': | |||
Revision as of 05:11, 10 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} }
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 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,
=
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 :
