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 Quote by stunner5000pt Oh i have another question about this Is this always true?? $$\hat{H} \Psi = E \Psi$$ Here we talk about the wavefunction in general I think this is not always true ... but i dont know understand why?? I m thinking that it ahs something to do with the time dependant case where the right is $$i\hbar \frac{d}{dt} \Psi$$ this term need not repsent the energy... right?
You've already been explained the difference between a time-dependent Hamiltonian in Schroedinger picture and a time-independent one. And how that determins the time-evolution of the state vector.

$$\hat{H} \Psi = E \Psi$$

is a spectral equation for an operator on separable Hilbert space and nothing more. It has solutions in the Hilbert space iff the spectrum of the Hamiltonian is discreet.

Daniel.