Adding angular momentum

  • Thread starter Jim Kata
  • Start date
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Say I wanted to tensor [tex]|1,-1> \otimes |1,0> [/tex] Then looking at the Clebsch Gordons I get [tex] |1,-1> \otimes |1,0> = \frac{1}{\sqrt {2}}|2,-1> - \frac{1}{\sqrt{2}}|1,-1>[/tex]
When I try to do this another way I run into a problem that I don't understand.

[tex] |1,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}> \otimes |\frac{1}{2},-\frac{1}{2}> + |\frac{1}{2},-\frac{1}{2}> \otimes |\frac{1}{2},\frac{1}{2}>) [/tex]

So

[tex]|1,-1> \otimes |1,0> = |1,-1> \otimes ( \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}> \otimes |\frac{1}{2},-\frac{1}{2}> + |\frac{1}{2},-\frac{1}{2}> \otimes |\frac{1}{2},\frac{1}{2}>))[/tex]

Going through this I get

[tex]|1,-1>\otimes|\frac{1}{2},\frac{1}{2}>\otimes|\frac{1}{2},-\frac{1}{2}>
=(\frac{1}{\sqrt{3}}|\frac{3}{2},-\frac{1}{2}> -\sqrt{\frac{2}{3}}|\frac{1}{2},-\frac{1}{2}>)\otimes|\frac{1}{2},-\frac{1}{2}>=\frac{1}{2}|2,-1> +(\frac{\sqrt{3}}{6}-\sqrt{\frac{2}{3}})|1,-1> [/tex]

similarly I get

[tex] |1,-1>\otimes|\frac{1}{2},-\frac{1}{2}>\otimes|\frac{1}{2},\frac{1}{2}>=\frac{\sqrt{3}}{2}|2,-1>-\frac{1}{2}|1,-1>[/tex]

but when I add these two I don't get

[tex] |1,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2}, \frac{1}{2}> \otimes |\frac{1}{2},-\frac{1}{2}> + |\frac{1}{2},-\frac{1}{2}> \otimes |\frac{1}{2},\frac{1}{2}>) [/tex]

What am I doing wrong? Where does my reasoning break down?
 

DrClaude

Mentor
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similarly I get

[tex] |1,-1>\otimes|\frac{1}{2},-\frac{1}{2}>\otimes|\frac{1}{2},\frac{1}{2}>=\frac{\sqrt{3}}{2}|2,-1>-\frac{1}{2}|1,-1>[/tex]
That's incorrect,
$$
|1,-1 \rangle \otimes|\frac{1}{2},-\frac{1}{2}\rangle\otimes|\frac{1}{2},\frac{1}{2}\rangle= \frac{1}{2} |2,-1\rangle - \frac{\sqrt{3}}{2}|1,-1 \rangle
$$
 

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