[Physics] Evaluate Commutator with Partial Derivatives

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I need to evaluate the following commutator…

$[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$

i tried applying an arbitrary function $\psi(x)$ but I am unsure of how to evaluate the whole thing. My first step was:

$=(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))*(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))\psi(x)-(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))*(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))\psi(x)$

This is where i am lost, I am unsure of where to go from here. Maybe there is something else I can do as my first step to simplify the question? please let me know what I can do. Thanks.

Best Answer

You can simplify your work using the identities

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

and

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

These are easy to prove by writing out the definition of a commutator.

This seems like it will give a lot of different terms, but most of them are zero. The $x,y,z$ all commute with each other, as do the derivatives. The only non-zero commutators are $[x,\frac{\partial}{\partial x}]$ and likewise for $y$ and $z$.

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