We are used to stationary Schrödinger equation. Slowly varying potential makes it more complicated. Physics should smoothen rapidly varying potential ... butlet us discuss what's happening while theoretical rapid change of potential(no adiabatic approximation).

For example imagine that potential has one minimum before the switch moment and a different one after (e.g. capacitor charged in one way then in opposite one) - like in this picture:

In minus infinity electron should be in the ground state of one potential and in plus infinity in ground state of the other -the question is how the transition of wavefunction would look like?

The main problem is thatquantum mechanics is time symmetric- such transition shouldn't be instant, so this symmetry suggests that the middle of this transition is the switch moment ... but it means thatthe wavefunction has started evolving before the switch???

I have to admit I don't understand the situation from perspective of quantum mechanics.

The above picture used Maximal Entropy Random Walk instead (page 48 of http://arxiv.org/abs/1111.2253 ) - corrected Brownian motion to finally become thermodynamical model - not only approximate maximum uncertainty principle, but really maximizing entropy. Thanks of it, it doesn't longer disagree with thermodynamical predictions of quantum mechanics - the equilibrium dynamical state has probability density being exactly the squares of the lowest energy eigenfunction of Schrödinger's Hamiltonian.

So this model agrees with the ground state in plus/minus infinities, but also naturally explains the transition - it indeed starts before the switch, but this time there is nothing strange about it: this model is thermodynamical - not fundamental but effective: we already know the history of potential and it allows us to estimate the best probability distribution of the particle. For example knowing that later it will be in another potential well allows us to tell that earlier it should be nearby.

There is another strange thing about above picture - its Ehrenfest Newton's equation has opposite sign - the particle accelerates uphill then decelerates downhill ... but it's just required e.g. to transport density between these minimums ...

But what's going on here in standard quantum mechanics?

Would the wavefunction start transforming in the moment of change or before?