# Vector calculus: how to order terms?

1. Oct 25, 2013

### 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

2. Oct 28, 2013

### 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...