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pGR0877 Problem 95

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (D) As you can check for yourself [J_x, J_y] = i \hbar J_z (the other two commutation relations can be obtain be cyclic permutation of the indexes x \to y, y \to z, z \to x; note also that I do not use the sign \hat{} for the operators). Using a sweet relation one must remember (or derive if needed) on the PGRE [AB,C] = A[B,C] + [A,C]B one has [J_xJ_y, J_x] = J_x[J_y, J_x] + [J_x, J_x]J_y = -i \hbar J_xJ_z (here I’ve used the property that any operator commutes with itself).

Found a typo? Comment!

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  1. Brent
    23.03.2012 at 20:16

    If you can’t remember the sweet relation, it can easily be derived:
    [AB,C] = ABC - CAB
    By adding and subtracting ACB we can expand this (adding and subtracting overall does not change the expression)
    [AB,C] = ABC - CAB + ACB - ACB = ABC - ACB + ACB - CAB = A(BC - CB) + (AC - CA)B = A[B,C] + [A,C]B

  2. darrin
    22.01.2015 at 01:11

    You have a typo. You have (Lx, Lx) should be (Lx, Lz)

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