Checking a commutation relation for angular momentum and lin. momentum

1. May 19, 2013

tomwilliam2

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

Prove the commutation relation $\left [L_x, p_y \right] = i\hbar \epsilon_{xyz} p_z$

2. Relevant equations
$L_x = yp_z - zp_y$
$p_z = i\hbar \frac{\partial}{\partial z}$

3. The attempt at a solution

$\left [L_x, p_y \right] = (yp_z - zp_y)p_y - p_y(yp_z - zp_y)$
$\left [L_x, p_y \right] = \left (y\cdot i\hbar \frac{\partial}{\partial z}\cdot i\hbar \frac{\partial}{\partial y} - z\cdot i\hbar \frac{\partial}{\partial y}\cdot i\hbar \frac{\partial}{\partial y}\right ) -\left ( i\hbar \frac{\partial y}{\partial y}i\hbar \frac{\partial}{\partial z} - i\hbar \frac{\partial z}{\partial y} i\hbar \frac{\partial}{\partial y} \right )$
$\left [L_x, p_y \right] = -y\hbar^2 \frac{\partial^2}{\partial z \partial y} + z\hbar^2 \frac{\partial^2}{\partial y} + \hbar^2\frac{\partial^2}{\partial z^2}$

I don't see how I can turn this into the final answer, given that I have derivatives of y still left...so where did I go wrong?

2. May 19, 2013

hexidecimel10

You need to remember to use the product rule when taking derivatives.

3. May 19, 2013

tomwilliam2

Thanks! That old mistake...