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I understand that we use [itex]i\hbar\partial/\partial t[/itex] and [itex]-i\hbar\nabla[/itex] for the energy and momentum operator but I would like to know how this identification is made.

I can see that it works for a wave of the form [itex]e^{i(kx-\omega t)}[/itex] and using the relation [itex]E=\hbar\omega[/itex] and the relation [itex]p=h/\lambda[/itex].

But what about everything else? How did physicists come to the conclusion that the general energy and momentum operator are of the form mentioned above?

Thank you very much.

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# Energy and momentum operators in QM

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