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Homework Help: Matrix represntation of angular momentum operator (QM)

  1. Feb 1, 2009 #1
    1. The problem statement, all variables and given/known data
    The matrix R(q) for rotating an ordinary vector by q around the z-axis is given by@

    cosq -sinq 0
    sinq cosq 0
    0 0 1

    From R calculate the matrix J(z).

    2. Relevant equations

    3. The attempt at a solution

    All I know is that U(q) = exp[-iJ(z)q] is the unitary operator which rotates a system, and I believe that

    R(q) x = U(dagger)(q) x U(q)

    Where x is the position vector.

    I have no idea where to go from here, other than expanding U with a taylor series but this didn't seem to go anywhere.

  2. jcsd
  3. Feb 1, 2009 #2


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    You will probably find page 315 and 316 of Shankar's Principles of Quantum Mechanic helpful.
  4. Feb 2, 2009 #3
    Thanks, but I still don't see how to do the question at all. Are the pages right - do they change with editions?
  5. Feb 2, 2009 #4


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    My edition of Shankar is from 1981. By chance I was reviewing this topic the day before your original post. I'll spend some time thinking about it and see if I can help you out.
  6. Feb 2, 2009 #5


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    I think that the correct relation is simply R(q) = exp[-i Jz q].

    Just work to first order in q. Then [itex] e^{-iJ_z q} \approx 1 - i J_z q [/itex] (where 1 here is the unit matrix) . Replace cos(q) by 1 and sin(q) by q in the matrix R(q) and you will get the expression for Jz easily.
  7. Feb 2, 2009 #6


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    I just logged on to make similar comments. Take the angle q to be very small, or infinitesimal, leads to the replacements given above.
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