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Difficulty with Sakurai, Ch.1, Problem # 9

  1. Dec 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the eigenvector for (S \cdot \hat{n}) |eigenvector> = (\hbar)/2 |eigenvector> where S = (\hbar)/2 \sigma. The sigmas are the Pauli spin matrices and \hat{n} = sin\beta cos\alpha \hat{i} + sin\beta\ sin\alpha \hat{j} + cos\beta \hat{k}

    You have to solve for the coefficients a and b of

    |eigenvector> = a|+> + b|->.

    2. The attempt at a solution

    It seems like when I try to solve for one coefficient, say a, is seems to vanish.

    a(\cos\beta - 1) + b\sin\beta e^{-i\alpha} = 0
    a\sin\beta e^{i\alpha} - b(\cos\beta + 1) = 0

    For example, isolating b in the 1st equation,

    b = - \frac{a(\cos\beta - 1)}{ \sin\beta e^{-i\alpha}}.

    If you plug this in the 2nd equation you get a factor of a for all terms, therefore the a's cancel each other.

    But Sakurai's solution is |eigenvector> = cos(\beta / 2)|+> + sin{\beta / 2} e^{i\alpha}|->. How can I get this?
     
  2. jcsd
  3. Dec 19, 2008 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi quantumkiko! :smile:

    (have an alpha: α and a beta: β and a sigma: σ :wink:)
    hmm … i can see σ3 in there somewhere …

    with eigenvectors which would be (a,b) = (1,0) or (0,1) if β = 0 or π …

    but whatever happened to poor little σ1 and σ2? :redface:

    (btw you have to put [noparse][tex] before and [/tex] after for the LaTeX to work in this forum[/noparse] :wink:)
     
  4. Dec 23, 2008 #3
    Thank you very much! ^^
     
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