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Coupled angular momentum

  1. Apr 8, 2007 #1
    1. The problem statement, all variables and given/known data

    say the coupled angular momentum eigenstate for two p elections is [tex]|2211>[/tex]. I am asked to operate on the eigenstate with [tex]L^2[/tex] and [tex]L_{z}[/tex] to verify [tex]lm[/tex]

    2. Relevant equations

    [tex]L^2 = L_1^2 +L_2^2 +2L_{1z}L_{2z}+(L_{1+}L_{2-}+L_{1-}L_{2+})[/tex]

    [tex]L_{+} |l,m>=|l,m+1>[/tex]
    [tex]L_{-} |l,m>=|l,m-1>[/tex]

    Eigenvalues:
    [tex]L^2=\hbar^2 l(l+1)[/tex]
    [tex]L_z=\hbar m[/tex]


    3. The attempt at a solution

    Using:

    [tex]L^2 = L_1^2 +L_2^2 +2L_{1z}L_{2z}+(L_{1+}L_{2-}+L_{1-}L_{2+})|2211>[/tex]

    I got it down to:

    [tex]2 \hbar |11>|11>+2 \hbar |11>|11>+2 \hbar |11>|11>+|12>|10>+|10>|12>[/tex]

    I guess my question is how to calculate |10>|12> and |12>|10> and if |10>|12>=|12>|10>=0 and why?

    Also, I was wondering why [tex]|2211>=C_{11}|11>|11>[/tex] and not [tex]|2211>=C_{02}|10>|12>+C_{20}|12>|10>+C_{12}|11>|11>[/tex]

    thanks,
    ux
     
    Last edited: Apr 8, 2007
  2. jcsd
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