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Quantum Mechanics - Spin

  1. Nov 14, 2007 #1
    [SOLVED] Quantum Mechanics - Spin

    1. The problem statement, all variables and given/known data
    Problem is attached.



    2. Relevant equations

    3. The attempt at a solution

    The first part is seemingly straight forward. Measurements are +/- hbar/2, both with probability (1/sqrt[2])^2 = 1/2 of being observed.

    For the next part I have written the operator S_x as:

    S_x = 1/2 [S+ + S-]
    Where S+ and S- are the raising and lowering operators respectively.
    ie. S+ = S_x + i S_y and S- = S_x - i S_y
    Then using:

    S+ | s,m > = [s(s+1 - m(m+1)]^(1/2) hbar | s,m+1 >
    S- | s,m > = [s(s+1 - m(m-1)]^(1/2) hbar | s,m+1 >

    With s = 1/2 (this is clear from the first part, since we have the eigenvalues of S_z (m) = -1/2 and 1/2 and -s <= m <= s in integer steps.

    I obtain
    S+ | 1/2 > = 0
    S- | 1/2 > = hbar | -1/2 >
    S+ | -1/2 > = hbar | 1/2 >
    S- | - 1/2 > = 0

    So
    S_x | 1/2 > = hbar/2 | -1/2 >
    S_x | -1/2 > = hbar/2 | 1/2 >

    and
    S_x | psi > = hbar/2 1/sqrt[2] ( | 1/2 > + | -1/2 > )
    So the measurement is simply hbar/2

    For the final part (where I become stuck!)
    S_y | 1/2 > = 1/2i (S+ - S-) | 1/2 > = ihbar/2 | -1/2 >
    S_y | -1/2 > = 1/2i (S+ - S-) | 1/2 > = -ihbar/2 | 1/2 >

    So I am required to find a state vector psi such that:
    S_y | psi > = -hbar/2 | psi >
    1/2i (S+ - S-) | psi > = -hbar/2 | psi >
    (S+ - S-) | psi > = -ihbar | psi >

    But is it even possible to construct a state vector out of the spin-up and spin-down eigenvectors to give this result? I can't seem to do it???
     

    Attached Files:

    • Q4.png
      Q4.png
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  2. jcsd
  3. Nov 14, 2007 #2

    nrqed

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    The attachment has been approved yet so I did not see the question but what you did all seems correct (disclaimer: I did not check all the coefficients but it all looks reasonable). For Sy, here's a trick: simply write psi as c_1 |+1/2> + c_2 |-1/2> and just impose that this be an eigenstate of S_y. That's all that is needed!
     
  4. Nov 14, 2007 #3
    So I should have (on the RHS)
    -hbar/2 [c_1 | 1/2 > + c_2 | -1/2 >]
    When I'm done, right?
    Or should I have:
    -hbar/2 [ | 1/2 > + | -1/2 >]

    I'm pretty sure it's the first one, right?

    Edit: Maybe not, I'm confusing myself with random coefficients multiplied for our cause and normalisation coefficients; aren't I?

    In which case I obtain c_1 = i and c_2 = -i

    Here is the question: http://img88.imageshack.us/img88/7308/q4rf5.png

    Edit2: Solved, c_1 = 1 and c_2 = -i. Just the eigenvector of the Pauli spin matrix sigma_y, duh!
     
    Last edited: Nov 15, 2007
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