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Euler angles. Quantum Mechanics Question

  1. Feb 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Let

    U = [itex]e^{iG_{3}\alpha}[/itex][itex]e^{iG_{2}\beta}[/itex][itex]e^{iG_{3}\gamma}[/itex]

    where ( [itex] \alpha, \beta, \gamma [/itex] ) are the Eulerian angles. In order that U represent a rotation ( [itex] \alpha, \beta, \gamma [/itex] ) , what are the commutation rules satisfied by the [itex] G_{k} [/itex] ?? Relate G to the angular momentum operators.

    2. Relevant equations



    3. The attempt at a solution

    I attached here the solution that i saw in my solution manual.. My question is how did he get

    [itex]G_{i}[/itex] = [itex]\frac{J_{i}}{\hbar}[/itex]

    I think it should be [itex]G_{i}[/itex] = [itex]{J_{i}}{\hbar}[/itex]

    Can someone please help me understand that solution?? Thanks. Help much appreciated.
     

    Attached Files:

    Last edited: Feb 3, 2012
  2. jcsd
  3. Feb 3, 2012 #2
    it should be one over h bar, and it is arisen from the commutation relations that result from infinitesimal rotations in 3 dimensional space. for more details please refer to advanced quantum mechanics texts as the proof is rather lengthy. i recommend reading:
    http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf
    specifically page 4, equations 9 to 12.
     
  4. Apr 8, 2013 #3
    hi there. i have been working on this problem recently, but i seem to have a slightly different answer to the one above. my working out led me to have a minus sign in the relation between G and J:

    after taking the taylor expansion of the exponentials and relating the [itex]\epsilon^{2}[/itex] coefficients i got:

    [itex]i^{2}G_{1}G_{2}-i^{2}G_{2}G_{1}=iG_{3}[/itex]
    so that [[itex]G_{2}, G_{1}[/itex]][itex]=iG_{3}[/itex]
    or [[itex]G_{1}, G_{2}[/itex]][itex] =-iG_{3}[/itex]

    this gave me [[itex]G_{i}, G_{j}[/itex]][itex]=-i\epsilon_{ijk}G_{k}[/itex]

    and hence i came up with the relation

    [itex]G_{i}=-J_{i}/\hbar[/itex]

    is this difference just some use of a different convention, or am i doing something wrong along the way??

    cheers
     
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