1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Anguluar momentum Commutation Identity

  1. Mar 12, 2015 #1
    1. The problem statement, all variables and given/known data

    Given that [itex][A_i,J_j]=i\hbar\epsilon_{ijk}Ak[/itex] where A_i is not invariant under rotation

    Show that [itex][J^2,Ai]=-2i\hbar\epsilon_{ijk}J_jAk-2\hbar^2A_i[/itex]

    2. Relevant equations
    [itex][AB,C]=A[B,C]+[A,C]B[/itex]

    [itex][A,B]=-[B,A][/itex]


    3. The attempt at a solution

    [itex][/itex]
    [itex][J^2,Ai]=[J_x^2,Ai]+[J_y^2,Ai]+[J_z^2,Ai][/itex]
    [itex]=J_x[J_x,Ai]+[J_x,Ai]J_x+J_y[J_y,Ai]+[J_y,Ai]J_y+J_z[J_z,Ai]+[J_z,Ai]J_z[/itex]
    [itex]=-J_x\epsilon_{ixk}Ak-\epsilon_{ixk}AkJ_x-J_y\epsilon_{iyk}Ak-\epsilon_{iyk}AkJ_y-J_z\epsilon_{izk}Ak-\epsilon_{izk}AkJ_z[/itex]

    Not sure where to go from here
     
  2. jcsd
  3. Mar 12, 2015 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    You left out a factor of ##i\hbar## in getting to your last line.

    Note that your final line can be written compactly as ##-\epsilon_{ijk}(J_jA_k + A_kJ_j)##
     
  4. Mar 12, 2015 #3
    Thanks that looks a lot easier to deal with, I guess I use that ##A_kJ_j= J_jA_k-[A_k,J_j]##?
     
  5. Mar 12, 2015 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    That's the right idea, but there's a sign error in your expression ##A_kJ_j= J_jA_k-[A_k,J_j]##.
     
  6. Mar 12, 2015 #5
    I have the exact same thing written?
     
    Last edited: Mar 12, 2015
  7. Mar 12, 2015 #6

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, I was just rewriting the same expression that you wrote. But the expression is incorrect due to sign errors.
     
  8. Mar 12, 2015 #7
    Ah right sorry, I had the right expression on the page and it worked so I was confused why you were correcting me
     
  9. Mar 12, 2015 #8
    To prove the full identity ##[J^2,[J^2,A]]=2\hbar^2(J^2A+AJ^2)-4\hbar^2(A\cdot J)J## can I just use a nested expression of what I just proved, as in let ##[J^2,Ai]=Ai## in my original identity
     
  10. Mar 12, 2015 #9

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, but of course ##[J^2,Ai] \neq Ai##. I managed to get the result, but only after a couple of pages of tedious index manipulations. I suspect there is a more elegant way to get to the result, but I don't see it.
     
  11. Mar 12, 2015 #10
    Its a bonus problem, and the last bonus problem was about 6 pages of tedious trigonometric stuff so it wouldn't surprise me if there wasn't, the identity is from dirac but I can't find his original derivation. Thanks for your help though
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Anguluar momentum Commutation Identity
  1. Commutator Identity (Replies: 2)

Loading...