Phy5646/CG coeff example2

We are to add angular momentum 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}= 1} 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}=1} to 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 j = 2,1,0} states. Using either the Ladder operator or the recursion relation, express all(nine) {j,m} eigenkets 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 |j_1 j_2; m_1 m_2\rangle} . Wrtie the answers as:

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, m=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 \frac{1}{\sqrt{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 \frac{1}{\sqrt{2}}|0,+\rangle} .......

where + and 0 stand 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 m_{1,2}=1,0} respectively. (Sakurai-3.20)

Answer:

We want to add the angular momentum 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}= 1} 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}= 1} to 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 j= |j_{1}-j_{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 j_{1}+j_{2}} = 0,1,2 states. Lets start from simplest 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 j= 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 m= 2} . This state is related 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 |j_1 m_1; j_2m_2\rangle} through the following equation:

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} = 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 \sum _{m= m_{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 \langle j_1 j_2; m_1m_2||j_1 j_2; jm\rangle |j_1 j_2; m_1m_2\rangle}

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} , 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= 2} in above equation, we get

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 m=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 \langle j_1 j_2; ++||j_1 j_2; jm\rangle |j_1 j_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 |++\rangle}

Now applying the 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 this state we get,

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_-|j=2;m=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-)|++\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_-|j=2;m=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{4}|j=2;m=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_1- + J_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{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 \sqrt{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=2;m=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{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 \sqrt{\frac{1}{2}}|+0\rangle}

Now similarly 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 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 |j=2;m=1\rangle} we get,

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_-|j=2;m=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{1}{2}}(J_1- + J_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 \sqrt{\frac{1}{2}}(J_1- + J_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 \Rightarrow } 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{6}|j=2;m=0\rangle} = 2Failed 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 |00\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 |-+\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 |+-\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=2;m=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 \sqrt{\frac{2}{3}}|00\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}{6}}|-+\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}{6}}|+-\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_-|j=2;m=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 \sqrt{\frac{2}{3}}(J_1- + J_2-)|00\rangle} + ${\sqrt {\frac {1}{6}}}(J_{1}-+J_{2}-)|+-\rangle$ + ${\sqrt {\frac {1}{6}}}(J_{1}-+J_{2}-)|-+\rangle$

$\Rightarrow$ ${\sqrt {6}}|j=2;m=-1\rangle$ = ${\sqrt {\frac {2}{3}}}{\sqrt {2}}|-0\rangle$ + ${\sqrt {\frac {2}{3}}}{\sqrt {2}}|0-\rangle$ + ${\sqrt {\frac {1}{6}}}{\sqrt {2}}|0-\rangle$ + ${\sqrt {\frac {1}{6}}}{\sqrt {2}}|-0\rangle$

$|j=2;m=-1\rangle$ = ${\sqrt {2}}|0-\rangle$ + ${\sqrt {2}}|-0\rangle$

$J_{-}|j=2;m=-1\rangle$ = ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|0-\rangle$ + ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|-0\rangle$

$\Rightarrow$ ${\sqrt {4}}|j=2;m=-2\rangle$ = ${\sqrt {\frac {1}{2}}}{\sqrt {2}}|--\rangle$ + ${\sqrt {\frac {1}{2}}}{\sqrt {2}}|--\rangle$

$|j=2;m=-2\rangle$ = $|--\rangle$

Now if we plug-in j=1 and m=1 in the first equation, we will get,

$|j=1;m=1\rangle$ = $a|+0\rangle$ + $b|0+\rangle$

This state should be orthogonal to all other $|jm\rangle$ states and in particular to $|j=2;m=1\rangle$ . Therefore,

$\langle j=2;m=1|j=1;m=1\rangle$ = 0

$a+b=0$ $\Rightarrow$ $a=-b$

Now we normalize the state $|j=1;m=1\rangle$ ,

$\langle j=1;m=1|j=1;m=1\rangle$ = 1 $\Rightarrow$ $|a|^{2}+|b|^{2}$ =1

$\Rightarrow$ $|a|={\sqrt {\frac {1}{2}}}$

$|j=1;m=1\rangle$ = ${\sqrt {\frac {1}{2}}}|+0\rangle$ - ${\sqrt {\frac {1}{2}}}|0+\rangle$

Now following the same procedure as above,

$J_{-}|j=1;m=1\rangle$ = ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|+0\rangle$ - ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|0+\rangle$

$|j=1;m=0\rangle$ = ${\sqrt {\frac {1}{2}}}|+-\rangle$ - ${\sqrt {\frac {1}{2}}}|-+\rangle$

Again, we can write,

$J_{-}|j=1;m=0\rangle$ = ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|+-\rangle$ - ${\sqrt {\frac {1}{2}}}(J_{1}-+J_{2}-)|+-\rangle$

$|j=1;m=-1\rangle$ = ${\sqrt {\frac {1}{2}}}|0-\rangle$ - ${\sqrt {\frac {1}{2}}}|-0\rangle$

Now if we start with the state $|j=0;m=0\rangle$ , it can be written as,

$|j=0;m=0\rangle$ = $c_{1}|00\rangle$ + $c_{2}|+-\rangle$ + $c_{3}|+-\rangle$

This state should be orthogonal to all other $|jm\rangle$ states and in particular to $|j=2;m=0\rangle$ . Therefore,

$\langle j=2;m=0|j=0;m=0\rangle$ = 0

$2c_{1}+c_{2}+c_{3}=0$

$\langle j=1;m=0|j=0;m=0\rangle$ = 0

$c_{2}=c_{3}$

Therefore from the last equations we get,

$c_{1}+c_{2}=0$ $\Rightarrow$ $c_{1}=-c_{2}$

This state $|j=0;m=0\rangle$ must be normalized

$\langle j=0;m=0|j=0;m=0\rangle$ = 1 $\Rightarrow$ $|c_{1}|^{2}+|c_{2}|^{2}+|c_{3}|^{2}$ =1

$|c_{2}|={\sqrt {\frac {1}{3}}}$

therefore, $c_{2}=c_{3}={\sqrt {\frac {1}{3}}}$ and $|c_{1}|=-{\sqrt {\frac {1}{3}}}$ . So we get

$|j=0;m=0\rangle$ = ${\sqrt {\frac {1}{3}}}|+-\rangle$ + ${\sqrt {\frac {1}{3}}}|-+\rangle$ - ${\sqrt {\frac {1}{3}}}|00\rangle$

Now we express the $|j;m\rangle$ state is terms of $|j_{1}j_{2};m_{1}m_{2}$ states or $|m_{1}m_{2}\rangle$ states.

$|j=2m=2\rangle$ = $|++\rangle$

$|j=2;m=1\rangle$ = ${\sqrt {\frac {1}{2}}}|0+\rangle$ + ${\sqrt {\frac {1}{2}}}|+0\rangle$

$|j=2;m=0\rangle$ = ${\sqrt {\frac {2}{3}}}|00\rangle$ + ${\sqrt {\frac {1}{6}}}|-+\rangle$ + ${\sqrt {\frac {1}{6}}}|+-\rangle$