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Checking a commutation relation for angular momentum and lin. momentum

  1. May 19, 2013 #1
    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?
    Thanks in advance
     
  2. jcsd
  3. May 19, 2013 #2
    You need to remember to use the product rule when taking derivatives.
     
  4. May 19, 2013 #3
    Thanks! That old mistake...
     
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