# Homework Help: Commutators of angular momentum

1. Jan 24, 2009

### SunGod87

1. The problem statement, all variables and given/known data
Show the three components of angular momentum: L_x, L_y and L_z commute with nabla^2 and r^2 = x^2 + y^2 = z^2

2. Relevant equations
[A, B] = AB - BA
For example:
$[L_x, \nabla^2] = L_x \nabla^2 - \nabla^2 L_x$

3. The attempt at a solution
$L_x \nabla^2 = -i\hbar(y\frac{\partial}{\partial z} \nabla^2 - z \frac{\partial}{\partial y}\nabla^2)$

$\nabla^2 L_x = -i\hbar\nabla^2(y\frac{\partial}{\partial z} - z \frac{\partial}{\partial y})$

How can I simplify these?

2. Jan 24, 2009

### Hurkyl

Staff Emeritus
Re: Commutators

If you have no better ideas, then why not rewrite $\nabla^2$, just like you did with $L_x$?

3. Jan 24, 2009

### snapback

Re: Commutators

after rewriting $\nabla^2$ as Hurkyl suggested, you should apply the commutator (it is also an operator) to a differentiable function $f(x,y,z)$ and see whether the differentiation rules (e.g.differentiation of a product) make the equation look simplier.