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Adding angular momentum

  1. Aug 27, 2011 #1
    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?
     
  2. jcsd
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