# Relating the Laplacian to the Quantum Angular Momentum

1. Sep 13, 2008

### davedude

This is my first post, so if it belongs somewhere else, please help me out. I've got a homework problem that I believe I solved, but I'm not sure if I did it right. (It seems too easy this way.)

I am given that $$\vec{L}$$=-i($$\vec{r}$$X$$\nabla$$). I have to prove the relation that:

$$\nabla$$2=1/r2($$\partial$$/$$\partial$$r(r2$$\partial$$/$$\partial$$r)-L2).

To solve this, I did the following:

L2=$$\vec{L}$$$$\bullet$$$$\vec{L}$$=-i($$\vec{r}$$X$$\nabla$$)$$\bullet$$-i($$\vec{r}$$X$$\nabla$$)=-r2$$\nabla$$2+($$\vec{r}$$$$\bullet$$$$\nabla$$)($$\vec{r}$$$$\bullet$$$$\nabla$$)

($$\vec{r}$$$$\bullet$$$$\nabla$$)($$\vec{r}$$$$\bullet$$$$\nabla$$)-L2=r2$$\nabla$$2

which yields:

$$\nabla$$2=1/r2($$\partial$$/$$\partial$$r(r2$$\partial$$/$$\partial$$r)-L2).

I apologize for the bad formatting. I can't figure out how to fix it. The black dot is the dot product. Assume that the black dot, the del, and the partial d's should all be lowered.