Phy5646/CG coeff example2: Difference between revisions
(New page: We are to add angular momentum <math>j_{1}= 1</math> and <math>j_{2}=1</math> to form <math>j = 2,1,0</math> states. Using either the Ladder operator or the recursion relation, express all...) |
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Now applying the <math>J_-</math> to this state we get, | Now applying the <math>J_-</math> to this state we get, | ||
<math>J_-|j=2;m=2\rangle</math>= <math>(J_1- + J_2-)|++\rangle/math> | <math>J_-|j=2;m=2\rangle</math>= <math>(J_1- + J_2-)|++\rangle</math> | ||
<math>J_-|j=2;m=2\rangle</math>= <math>\sqrt{4}|j=2;m=1\rangle</math> ; <math>(J_1- + J_2-)|++\rangle/math>= <math>\sqrt{2}|0+\rangle/math> + <math>\sqrt{2}|+0\rangle/math> | <math>J_-|j=2;m=2\rangle</math>= <math>\sqrt{4}|j=2;m=1\rangle</math> ; <math>(J_1- + J_2-)|++\rangle</math>= <math>\sqrt{2}|0+\rangle</math> + <math>\sqrt{2}|+0\rangle</math> | ||
<math>|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0+\rangle/math> + <math>\sqrt{\frac{1}{2}}|+0\rangle/math> | <math>|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0+\rangle</math> + <math>\sqrt{\frac{1}{2}}|+0\rangle</math> | ||
Now similarly applying <math>J_-</math> to the state <math>|j=2;m=1\rangle</math> we get, | Now similarly applying <math>J_-</math> to the state <math>|j=2;m=1\rangle</math> we get, | ||
<math>J_-|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0+\rangle/math> + <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+0\rangle/math> | <math>J_-|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0+\rangle</math> + <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+0\rangle</math> | ||
<math>\Rightarrow </math> <math>\sqrt{6}|j=2;m=0\rangle</math> = 2<math>|00\rangle/math> + <math> |-+\rangle/math> + <math> |+-\rangle/math> | <math>\Rightarrow </math> <math>\sqrt{6}|j=2;m=0\rangle</math> = 2<math>|00\rangle</math> + <math> |-+\rangle</math> + <math> |+-\rangle</math> | ||
<math>|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}|00\rangle/math> + <math> \sqrt{\frac{1}{6}}|-+\rangle/math> + <math> \sqrt{\frac{1}{6}}|+-\rangle/math> | <math>|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}|00\rangle</math> + <math> \sqrt{\frac{1}{6}}|-+\rangle</math> + <math> \sqrt{\frac{1}{6}}|+-\rangle</math> | ||
<math>J_-|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}(J_1- + J_2-)|00\rangle/math> + <math>\sqrt{\frac{1}{6}}(J_1- + J_2-)|+-\rangle/math> +<math>\sqrt{\frac{1}{6}}(J_1- + J_2-)|-+\rangle/math> | <math>J_-|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}(J_1- + J_2-)|00\rangle</math> + <math>\sqrt{\frac{1}{6}}(J_1- + J_2-)|+-\rangle</math> +<math>\sqrt{\frac{1}{6}}(J_1- + J_2-)|-+\rangle</math> | ||
<math>\Rightarrow </math> <math>\sqrt{6}|j=2;m=-1\rangle</math> = <math>\sqrt{\frac{2}{3}}\sqrt{2}|-0\rangle/math> + <math>\sqrt{\frac{2}{3}}\sqrt{2}|0-\rangle/math> + <math>\sqrt{\frac{1}{6}}\sqrt{2}|0-\rangle/math> + <math>\sqrt{\frac{1}{6}}\sqrt{2}|-0\rangle/math> | <math>\Rightarrow </math> <math>\sqrt{6}|j=2;m=-1\rangle</math> = <math>\sqrt{\frac{2}{3}}\sqrt{2}|-0\rangle</math> + <math>\sqrt{\frac{2}{3}}\sqrt{2}|0-\rangle</math> + <math>\sqrt{\frac{1}{6}}\sqrt{2}|0-\rangle</math> + <math>\sqrt{\frac{1}{6}}\sqrt{2}|-0\rangle</math> | ||
<math>|j=2;m=-1\rangle</math> = <math>\sqrt{2}|0-\rangle/math> + <math>\sqrt{2}|-0\rangle/math> | <math>|j=2;m=-1\rangle</math> = <math>\sqrt{2}|0-\rangle</math> + <math>\sqrt{2}|-0\rangle</math> | ||
<math>J_-|j=2;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0-\rangle/math> + <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|-0\rangle/math> | <math>J_-|j=2;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0-\rangle</math> + <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|-0\rangle</math> | ||
<math>\Rightarrow </math> <math>\sqrt{4}|j=2;m=-2\rangle</math> = <math>\sqrt{\frac{1}{2}}\sqrt{2}|--\rangle/math> + <math>\sqrt{\frac{1}{2}}\sqrt{2}|--\rangle/math> | <math>\Rightarrow </math> <math>\sqrt{4}|j=2;m=-2\rangle</math> = <math>\sqrt{\frac{1}{2}}\sqrt{2}|--\rangle</math> + <math>\sqrt{\frac{1}{2}}\sqrt{2}|--\rangle</math> | ||
<math>|j=2;m=-2\rangle</math> = <math>|--\rangle/math> | <math>|j=2;m=-2\rangle</math> = <math>|--\rangle</math> | ||
Now if we plug-in j=1 and m=1 in the first equation, we will get, | Now if we plug-in j=1 and m=1 in the first equation, we will get, | ||
<math>|j=1;m=1\rangle</math> = <math>a|+0\rangle/math> + <math>b|0+\rangle/math> | <math>|j=1;m=1\rangle</math> = <math>a|+0\rangle</math> + <math>b|0+\rangle</math> | ||
This state should be orthogonal to all other <math>|jm\rangle</math> states and in particular to <math>|j=2;m=1\rangle</math>. Therefore, | This state should be orthogonal to all other <math>|jm\rangle</math> states and in particular to <math>|j=2;m=1\rangle</math>. Therefore, | ||
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<math>\Rightarrow </math> <math>|a|=\sqrt{\frac{1}{2}}</math> | <math>\Rightarrow </math> <math>|a|=\sqrt{\frac{1}{2}}</math> | ||
<math>|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|+0\rangle/math> - <math>\sqrt{\frac{1}{2}}|0+\rangle/math> | <math>|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|+0\rangle</math> - <math>\sqrt{\frac{1}{2}}|0+\rangle</math> | ||
Now following the same procedure as above, | Now following the same procedure as above, | ||
<math>J_-|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+0\rangle/math> - <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0+\rangle/math> | <math>J_-|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+0\rangle</math> - <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|0+\rangle</math> | ||
<math>|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}|+-\rangle/math> - <math>\sqrt{\frac{1}{2}}|-+\rangle/math> | <math>|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}|+-\rangle</math> - <math>\sqrt{\frac{1}{2}}|-+\rangle</math> | ||
Again, we can write, | Again, we can write, | ||
<math>J_-|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+-\rangle/math> - <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+-\rangle/math> | <math>J_-|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+-\rangle</math> - <math>\sqrt{\frac{1}{2}}(J_1- + J_2-)|+-\rangle</math> | ||
<math>|j=1;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0-\rangle/math> - <math>\sqrt{\frac{1}{2}}|-0\rangle/math> | <math>|j=1;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0-\rangle</math> - <math>\sqrt{\frac{1}{2}}|-0\rangle</math> | ||
Now if we start with the state <math>|j=0;m=0\rangle</math>, it can be written as, | Now if we start with the state <math>|j=0;m=0\rangle</math>, it can be written as, | ||
<math>|j=0;m=0\rangle</math> = <math>c_1|00\rangle/math> + <math>c_2|+-\rangle/math> + <math>c_3|+-\rangle/math> | <math>|j=0;m=0\rangle</math> = <math>c_1|00\rangle</math> + <math>c_2|+-\rangle</math> + <math>c_3|+-\rangle</math> | ||
This state should be orthogonal to all other <math>|jm\rangle</math> states and in particular to <math>|j=2;m=0\rangle</math>. Therefore, | This state should be orthogonal to all other <math>|jm\rangle</math> states and in particular to <math>|j=2;m=0\rangle</math>. Therefore, | ||
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Therefore from the last equations we get, | Therefore from the last equations we get, | ||
<math>c_1+c_2= 0</math> math>\Rightarrow </math> <math>c_1 = -c_2</math> | <math>c_1+c_2= 0</math> <math>\Rightarrow </math> <math>c_1 = -c_2</math> | ||
This state <math>|j=0;m=0\rangle</math> must be normalized | This state <math>|j=0;m=0\rangle</math> must be normalized | ||
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<math>|j=0;m=0\rangle</math> = <math>\sqrt{\frac{1}{3}}|+-\rangle/math> + <math>\sqrt{\frac{1}{3}}|-+\rangle/math> - <math>\sqrt{\frac{1}{3}}|00\rangle/math> | <math>|j=0;m=0\rangle</math> = <math>\sqrt{\frac{1}{3}}|+-\rangle</math> + <math>\sqrt{\frac{1}{3}}|-+\rangle</math> - <math>\sqrt{\frac{1}{3}}|00\rangle</math> | ||
Now we express the <math>|j;m\rangle</math> state is terms of <math>|j_1 j_2; m_1 m_2</math> states or <math>|m_1 m_2\rangle</math> states. | Now we express the <math>|j;m\rangle</math> state is terms of <math>|j_1 j_2; m_1 m_2</math> states or <math>|m_1 m_2\rangle</math> states. | ||
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<math>|j=2 m=2\rangle</math> = <math>|++\rangle</math> | <math>|j=2 m=2\rangle</math> = <math>|++\rangle</math> | ||
<math>|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0+\rangle/math> + <math>\sqrt{\frac{1}{2}}|+0\rangle/math> | <math>|j=2;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0+\rangle</math> + <math>\sqrt{\frac{1}{2}}|+0\rangle</math> | ||
<math>|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}|00\rangle/math> + <math> \sqrt{\frac{1}{6}}|-+\rangle/math> + <math> \sqrt{\frac{1}{6}}|+-\rangle/math> | <math>|j=2;m=0\rangle</math> = <math>\sqrt{\frac{2}{3}}|00\rangle</math> + <math> \sqrt{\frac{1}{6}}|-+\rangle</math> + <math> \sqrt{\frac{1}{6}}|+-\rangle</math> | ||
<math>|j=2;m=-1\rangle</math> = <math>\sqrt{2}|0-\rangle/math> + <math>\sqrt{2}|-0\rangle/math> | <math>|j=2;m=-1\rangle</math> = <math>\sqrt{2}|0-\rangle</math> + <math>\sqrt{2}|-0\rangle</math> | ||
<math>|j=2;m=-2\rangle</math> = <math>|--\rangle/math> | <math>|j=2;m=-2\rangle</math> = <math>|--\rangle</math> | ||
<math>|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|+0\rangle/math> - <math>\sqrt{\frac{1}{2}}|0+\rangle/math> | <math>|j=1;m=1\rangle</math> = <math>\sqrt{\frac{1}{2}}|+0\rangle</math> - <math>\sqrt{\frac{1}{2}}|0+\rangle</math> | ||
<math>|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}|+-\rangle/math> - <math>\sqrt{\frac{1}{2}}|-+\rangle/math> | <math>|j=1;m=0\rangle</math> = <math>\sqrt{\frac{1}{2}}|+-\rangle</math> - <math>\sqrt{\frac{1}{2}}|-+\rangle</math> | ||
<math>|j=1;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0-\rangle/math> - <math>\sqrt{\frac{1}{2}}|-0\rangle/math> | <math>|j=1;m=-1\rangle</math> = <math>\sqrt{\frac{1}{2}}|0-\rangle</math> - <math>\sqrt{\frac{1}{2}}|-0\rangle</math> | ||
<math>|j=0;m=0\rangle</math> = <math>\sqrt{\frac{1}{3}}|+-\rangle/math> + <math>\sqrt{\frac{1}{3}}|-+\rangle/math> - <math>\sqrt{\frac{1}{3}}|00\rangle/math> | <math>|j=0;m=0\rangle</math> = <math>\sqrt{\frac{1}{3}}|+-\rangle</math> + <math>\sqrt{\frac{1}{3}}|-+\rangle</math> - <math>\sqrt{\frac{1}{3}}|00\rangle</math> | ||
Latest revision as of 14:30, 27 April 2010
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} + +
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=-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{4}|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{\frac{1}{2}}\sqrt{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}{2}}\sqrt{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=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 |--\rangle}
Now if we plug-in j=1 and m=1 in the first equation, we will 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=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 a|+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 b|0+\rangle}
This state should be orthogonal to all other 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 |jm\rangle} states and in particular 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=2;m=1\rangle} . Therefore,
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=2;m=1|j=1;m=1\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 a+b= 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 \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 a= -b }
Now we normalize 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=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 \langle j=1;m=1|j=1;m=1\rangle} = 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 \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 |a|^{2} + |b|^{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 \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 |a|=\sqrt{\frac{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 |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 \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 following the same procedure as above,
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;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 |j=1;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{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}{2}}|-+\rangle}
Again, we can write,
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;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{1}{2}}(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{\frac{1}{2}}(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=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 \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 if we start with 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=0;m=0\rangle} , it can be written 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=0;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 c_1|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 c_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 c_3|+-\rangle}
This state should be orthogonal to all other 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 |jm\rangle} states and in particular 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=2;m=0\rangle} . Therefore,
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=2;m=0|j=0;m=0\rangle} = 0
= 0
Therefore from the last equations we get,
This state must be normalized
= 1 =1
therefore, and . So we get
= + -
Now we express the state is terms of states or states.
=
= +
= + +
= +
=
= -
= -
= -
= + -