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Could someone explain what this sentence mean? "If the Hamiltonian changes suddenly by a finite amount, the wavefunction must change continuously in order that the time-dependent Schrodinger equation be valid"?

I am guessing that such a change describes a square-well change? And is it saying that ${\partial \psi(x,t)\over\partial t}$ has to be continuous in such a case? Thanks.

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If Hamiltonian changes suddenly, the wave function time derivative does too, so it is not continuous. It is the wave function itself who is continuous: $\psi(t) = \int_{t_1}^t H\psi(t')dt'$ –  Vladimir Kalitvianski Sep 4 '11 at 19:37

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You are almost correct. Continuous $\psi(x,t)$ (as a function of time) is by definition wavefunction with finite time derivative. Time derivative is finite because Hamiltonian is finite (due to time-dependent SE).

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