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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?

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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.

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