Angular mometum and the commutator?

[cragar]

Homework Statement

show that
$$[L_z,L_x]=i(\hbar)L_y$$

The Attempt at a Solution

[A,B]=AB-BA
$$L_z=xP_y-yP_x$$
$$L_x=yP_z-ZP_x$$
So do i just use the fact that [A,B]= AB-BA
and then use the momentum operators and substitute everything in an churn out the algebra to reduce it down.

 

[gomboc]

You already know what the operators $$L_x,\ L_z$$ are defined as in terms of differentials.

For example, $$L_x = -i\hbar\left(y\frac{\partial}{\partial x} - z\frac{\partial}{\partial y}\right)$$

Just make use of the following operator identities:

$$[A,B+C] = [A,B]+[A,C]$$
$$[A,BC] = [A,B]C+B[A,C]$$

and you should be able to manipulate them into what you want. I'm not sure $$[A,B] = AB - BA$$ would have been sufficient in and of itself.

 

[cragar]

thanks for your response