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Quick angular momentum question

  1. Jan 22, 2006 #1
    A hopefully, quick issue I just wanted to be clear on before my exam.

    One question gave me two quantum numbers for total angular momentum.
    j1 = 1 and j2 = 2. The question asks to list the Eigenvalues for toatl angular momentum and its z component.

    I think I can do the later part okay by working out the possible combinations of the various m quantum number.

    However I'm a little unsure about the first part? Should I be calculating the Eigenvalues for the square of each angular momentum component j1^2 and j2^2, square rooting or adding (or adding an square rooting). Or should I add the numbers together then calculate the Eigenvalue?

    Or none of the above:redface:

    I hope that all made sense
     
  2. jcsd
  3. Jan 23, 2006 #2
    okay, when [tex]J=j_1+j_2[/tex], possible values of J are [|j_1-j_2|<J<j_1+j_2[/tex].
    eigenvalues and eigen vectors for [tex]J_z[/tex] are: [tex]J_z|J,M_J>=M_J|J,M_J>=(m_1+m_2)|J,M_J)[/tex]
    and eigenvalues for [tex]J^2[/tex] are [tex]J(J+1)[/tex] so just aqrt it for the total angular momentum.

    theres no need to decouple the angular momentums to find these values (and it isnt recommanded either), if youd try to get the values for the decoupled momentum youd have [tex](j_1^1+2j_1j_2+j_2^2)(|j_1,m_{j_1}>+|j_2,m_{j_2}>)[/tex] as you can see, youd have to diagonize the matrix inorder to get the eigen values, because [tex]j_1j_2=j_{1z}j_{2z}+j_{1x}j_{2x}+j_{1y}j_{2y}=j_{1z}j_{2z}+\frac{1}{2}(j_{1+}j_{2-}+j_{1-}j_{2+})[/tex]
     
    Last edited: Jan 23, 2006
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