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Vector calculus: how to order terms?

  1. Oct 25, 2013 #1
    Hi,

    I need to let an operator act on a scalar function. The operator is however in a very cryptic form, so I would want to work it out a little bit. I get stuck in the process. The operator is:

    [itex]\vec{u}\cdot\left[\vec{L}\times\left(\vec{u}_r\times\vec{L}\right)\right]f[/itex]

    Where [itex]\vec{L}[/itex] is the angular momentum operator and [itex]\vec{u}_r[/itex] is the unit vector along the [itex]r[/itex]-direction in a spherical coordinate system. [itex]\vec{u}[/itex] is just a constant vector.
    The outer [itex]\vec{L}[/itex] needs to work on both the [itex]\vec{u}_r[/itex] and the inner [itex]\vec{L}[/itex], so I write:

    [itex]\vec{u}\cdot\left[\check{\vec{u}_r}\left(\check{\vec{L}}\cdot\vec{L}\right)-\left(\check{\vec{L}} \cdot \check{\vec{u}_r}\right)\vec{L}+\vec{u}_r \left(\check{\vec{L}}\cdot \check{\vec{L}}\right)-\left(\vec{u}_r \cdot \check{\vec{L}}\right)\check{\vec{L}}\right]f[/itex][itex][/itex]

    where the upside-down hat denotes the vector on which the [itex]\check{\vec{L}}[/itex] operator acts. In the last three terms the ordering of the operators is correct.
    For the first term I do not see how to let [itex]\check{\vec{L}}[/itex] operate on [itex]\check{\vec{u}_r}[/itex] without the second [itex]\vec{L}[/itex] operator acting on it as well. My best guess is to make a tensor term of the sort

    [itex]\left(\check{\vec{L}}\check{\vec{u}_r}\right)\cdot\vec{L}[/itex]

    but I'm not sure about the ordering of the terms in this expression. I have checked the tensor and it is anti-symmetric, so the ordering will make a difference.

    Any suggestions are welcome, thanks in advance!

    Giorgos
     
  2. jcsd
  3. Oct 28, 2013 #2
    The problem is solved in the meanwhile. The result is

    [itex]\vec{u}\cdot\left[\left(\check{\vec{L}}\check{\vec{u}_r}\right)^T\cdot\vec{L}-\left(\check{\vec{L}} \cdot \check{\vec{u}_r}\right)\vec{L}+\vec{u}_r \left(\check{\vec{L}}\cdot \check{\vec{L}}\right)-\left(\vec{u}_r \cdot \check{\vec{L}}\right)\check{\vec{L}}\right]f[/itex]

    For people who would meet such problems in the future, I strongly advice you to use Mathematica to check the results. In this case it saved me...
     
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