QM: commutator 2D particle magnetic field

[zhaos]

Homework Statement

I was reading this textbook:
https://books.google.com/books?id=s...10#v=onepage&q=orbit center operators&f=false

Homework Equations

In the equation of the page (unlabeled), we have
$$
\left[A_x\frac{\partial}{\partial y} - \frac{\partial}{\partial y}A_x + \frac{\partial}{\partial x}A_y - A_y\frac{\partial}{\partial x}\right]\psi(x)\\
= \left[- \frac{\partial A_x}{\partial y}+ \frac{\partial A_y}{\partial x}\right]\psi(x)
$$

The Attempt at a Solution

Why do the first and fourth terms cancel out? Is it to do with the circular motion of the charged particle in the magnetic field?

 

[blue_leaf77]

For example, for the second term in the first line of your equation
$$
- \frac{\partial}{\partial y}(A_x \psi(x)) = - A_x \frac{\partial \psi(x)}{\partial y} - \psi(x)\frac{\partial A_x}{\partial y}
$$
The first term of the right hand side in the above equation will cancel with the first term in

$$
\left[A_x\frac{\partial}{\partial y} - \frac{\partial}{\partial y}A_x + \frac{\partial}{\partial x}A_y - A_y\frac{\partial}{\partial x}\right]\psi(x)
$$

 

[zhaos]

Oh. That makes it clear. Thank you.