How to evaluate commutator with angular momentum? I need to evaluate the commutator $[\hat{x},\hat{L}_z]$. I believe the $L_z$ is referring to the angular momentum operator which is:
$L_z = xp_y - yp_x$
using this relationship i end up with:
$[x,L_z] = x(xp_y - yp_x)-(xp_y - yp_x)x$
my next step is substituting in for the p operator but i still dont get anywhere. any suggestions???
 A: Usually I find it easiest to evaluate commutators without resorting to an explicit (position or momentum space) representation where the operators are represented by differential operators on a function space.
In order to evaluate commutators without these representations, we use the so-called canonical commutation relations (CCRs)
$$
  [x_i,p_j] = i\hbar \,\delta_{ij}, \qquad [x_i, x_j]=0,\qquad [p_i, p_j]=0
$$
Now, in order to evaluate and angular momentum commutator, we do precisely as you suggested using the expression
$$
  L_z = x p_y - y p_x
$$
and we use the CCRs
\begin{align}
  [x, L_z] &= [x, xp_y-yp_x]\\
&= [x,xp_y] - [x,yp_x]\\
&= x[x,p_y]+[x,x]p_y-y[x,p_x]-[x,y]p_x \\
&= -i\hbar y
\end{align}
In the last step, only the third term was non-vanishing because of the CCRs.  I have also used the fact that the commutator is linear in both of its arguments, 
$$
  [aA+bB,C] = a[A,C] + b[B,C], \qquad [A,bB + cC] = b[A,B] + c[A,C]
$$
where $a,b,c$ are numbers and $A,B,C$ are operators, and the following commutator identity that you'll find useful in general:
$$
  [AB,C] = A[B,C] + [A,C]B
$$
A: I'll help you with a general solution. First, you can write the angular momentum as $L_{i}=\epsilon_{ijk}x_{j}p_{k}$. Where $e_{ijk}$ is the Levi-Civita symbol. Now write a general commutator as $[x_{l},L_{i}]$ where $l,i$ run from 1 to 3. With this you have 
$[x_{l},L_{i}]=[x_{l},\epsilon_{ijk}x_{j}p_{k}]$
Now, you use the following identity for commutator $[A,BC]=[A,B]C+B[A,C]$. With this in mind, you get
$[x_{l},\epsilon_{ijk}x_{j}p_{k}]=\epsilon_{ijk}[x_{l},x_{j}]p_{k}+\epsilon_{ijk}x_{j}[x_{l},p_{k}]$
Now, the commutator $[x_{l},x_{j}]$ is zero and $[x_{l},p_{k}]$ is equal to $i\hbar\delta_{lk}$. So, you are left with 
$[x_{l},\epsilon_{ijk}x_{j}p_{k}]=[x_{l},L_{i}]=i\hbar\epsilon_{ijk}x_{j}\delta_{lk}=i\hbar\epsilon_{ijl}x_{j}$
From this result you can find your particular commutator.
