- #1
tomwilliam2
- 117
- 2
Homework Statement
Prove the commutation relation ##\left [L_x, p_y \right] = i\hbar \epsilon_{xyz} p_z##
Homework Equations
##L_x = yp_z - zp_y##
##p_z = i\hbar \frac{\partial}{\partial z}##
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?
Thanks in advance