# Physical meaning of $\langle nlm|\hat{z}|n'l'm'\rangle$

I'm working on a quantum mechanics problem with some friends and we're trying to make an argument using symmetry rather than maths. What would the physical interpretation of $$\langle nlm|\hat{z}|n'l'm'\rangle$$ be for two states of the Hydrogen atom?

The simplest interpretation would be as a transition amplitude. If an electric dipole oriented along the $$z$$ axis interacted with your atom, the probability of transition from $$\vert n'l'm'\rangle$$ to $$\vert nlm\rangle$$ would be proportional to \begin{align} \vert \langle nlm\vert \hat z \vert n'l'm'\rangle\vert^2\, . \end{align}
Note that the operator $$\hat z$$ can only connect $$n'$$ with $$n=n'\pm 1$$, $$l$$ with $$l'=l\pm 1$$ and $$m'=m$$. If the orientation was different there could be a change in the magnetic quantum numbers. The process $$n'\to n'+1$$ is clearly absorption whereas $$n'\to n'-1$$ is emission.