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Homework Help: Problem following a demonstration

  1. Aug 5, 2016 #1
    1. The problem statement, all variables and given/known data

    Good morning or afternoon everybody. I am trying to calculate the commutator between two components of angular momenta by following somebody's steps. During the process, I arrive to one step which I can't follow. It's between before and after the equal.

    Σnmljnmkln iħ XlPm - Σnmkjnmkmh iħ XnPh = iħ(Σml (-δjkXlPl)+ΣmjkmXjPkmkXnPnδjknjkmXjPk).

    What I don't get is how we transform the levi civita symbols (∈)

    2. Relevant equations


    3. The attempt at a solution

    In this case, we can transform the symbols of the left part of the equal to get four deltas; ∈jmnkln = -δjkδml + δjlδmk & ∈jnmkmh = -δjkδnh + δjhδmk.

    Then, we can use δml on XlPm and δmk on PmXl to obtain what is seen at the left of the equal, Σml (-δjkXlPl) & ΣmkXnPnδjk

    But I don't know how can I calculate the other two terms, ΣmjkmXjPk & -ΣnjkmXjPk.

    If anyone can tell me how to calculate them, I would be very thankfull
    Last edited by a moderator: Aug 6, 2016
  2. jcsd
  3. Aug 5, 2016 #2


    Staff: Mentor

    Can you check your indices again?

    It seems that the -Σn∈jkmXjPk has no n to index with the summation.

    The other ∈jnm∈kmh= -δjkδnh + δjhδmk. looks off too as the righthand side sums over m, right?
    and so there should not be an m index on the lefthand side.

    CAVEAT: I'm no expert so wait until a math advisor comments here (@micromass or @Mark44)
  4. Aug 6, 2016 #3
    You are right, it is ∈jkn.
    On the other hand, I wanted to post an image from wikipedia in which the product of two levi civita symbols is described, but I can't copy the image, what I want to do is to write the symbols as a matrix of deltas, then grab the δii multiplied by the substraction of two products of deltas.
  5. Aug 6, 2016 #4


    User Avatar

    Staff: Mentor

    It seems to not be a image. So you'd need to take a screenshot, then attach to your post that screenshot image.

    EDIT: If you surround wikipedia's SVG link with IMG tags, you could request that viewers click on Reply (whether or not they may intend replying), and they will see that the image does display as intended in the pop-up editor pane.

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