Vector calculus: how to order terms?

[giokara]

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:

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

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

\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

where the upside-down hat denotes the vector on which the \check{\vec{L}} 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 \check{\vec{L}} operate on \check{\vec{u}_r} without the second \vec{L} operator acting on it as well. My best guess is to make a tensor term of the sort

\left(\check{\vec{L}}\check{\vec{u}_r}\right)\cdot\vec{L}

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

 

[giokara]

The problem is solved in the meanwhile. The result is

\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

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...