While reading Sakurai's Modern QM, I was stuck at the point where he explains the absorption and emission of light quanta in atoms. He proceeds with Hamiltonian:
$$H= p^2/2m + e\phi(x) -e/mc A\cdot p$$ where $p$ is momentum, $m$ is mass of an electron, $A$ is magnetic potential.
Next, he says to drop terms $p\cdot A$ and $|A|^2$ terms. He says that since $\nabla \cdot A=0$ the term can be dropped. However, I do not see any reason to drop $|A|^2$ term. Can anyone justify this?
My next question was about expectation value of $A\cdot p$. Sakurai writes:
$$\langle x|p\cdot A|y\rangle = -ih\nabla\cdot (A(x)\langle x| \rangle) = \langle x|A\cdot p| \rangle + \langle x| \rangle [-ih\nabla\cdot A(x)]$$
Can anyone explain how he manipulated the above equation to arrive at the result?