I would appreciate if someone could set me straight here. I understand if I have an arbitrary operator, I can express it in matrix component notation as follows: Oi,j = <vi|O|vj> Is it possible to get a representation of the operator O back from this component form. I'm more interested in what to do in the case of continuous i and j so I assume I will have to do a 2 dimensional integral. Also how could I go from this form to calculate an expectation value <O>. Again I would have thought this would be a 2D integral as-well but something a bit more as I would want this <X|O|X> where X is a wave-function.