Homework Help: Curl and Angular Momentum

1. Sep 17, 2011

1. The problem statement, all variables and given/known data

I want to prove this: $\vec{r} \nabla^2 - \nabla(1+r \frac{\partial}{\partial {r}})=i \nabla \times \vec{L}$

2. Relevant equations

BAC-CAB rule:

$\nabla \times (\vec{A} \times \vec{B}) = (\vec{B} . \nabla) \vec{A} - (\vec{A} . \nabla ) \vec{B} - \vec{B} (\nabla . \vec{A} ) + \vec{A} (\nabla . \vec{B})$

3. The attempt at a solution

We know that $i \nabla \times \vec{L}= \nabla \times (\vec{r} \times \nabla)$.

Now how can I continue with $\nabla \times (\vec{r} \times\nabla) =$?

Can I use BAC-CAB rule? When I use BAC-CAB, I will have this: $\nabla \times (\vec{r} \times\nabla) = \vec{r} (\nabla . \nabla) - \nabla (\nabla . \vec{r}) + (\nabla . \nabla) \vec{r}- (\vec{r} . \nabla) \nabla$

And as we know $\nabla (\nabla . \vec{r}) = \vec{0}$.

Now how can I continue? If my method is wrong, please tell me why!

Last edited: Sep 17, 2011