[vertbhrtn]

Would it really just jump into another shape? I don't understand QM much, but the Schrodinger equation looks like a typical heat equation (with some complex-number trickery) and thus in general should evolve smoothly. There should be a transition period between two shapes.

[plus]

You are spot on here: the heat equation and the Schrodinger equation are both special cases of the wave equation. A major difference between the two is that the Schrodinger equation deals with complex numbers, as you state. Transition between states is not instantaneous, though it is very fast, on the scale of attoseconds (10^-18 seconds).