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We've got the Hamiltonian operator H acting on the state vector [tex] |\Psi\rangle[/tex] and it gives you the eigenvalue E which is what we can measure in experiments. That much is fair enough.

But, it also says that when you apply the operator, you are doing a time evolution on the state [tex] |\Psi\rangle[/tex]. I am struggling to see what the link is between energy and time evolution. I know I'm not asking a very precise question here, but what is the significance of this?

What does it mean to equate energy with time evolution? For that matter, what does it mean to equation momentum p with [tex] i\hbar\frac{\partial}{\partial x} [/tex],

("space evolution" I suppose you might call that?)

This feels like something basic I should be aware of by now.