QM: commutator 2D particle magnetic field

1. Nov 1, 2015

zhaos

1. The problem statement, all variables and given/known data
I was reading this textbook:

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

3. 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?

2. Nov 1, 2015

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

3. Nov 1, 2015

zhaos

Oh. That makes it clear. Thank you.