Well, the specification of probabilities and phases above contains all the information about the state, so it has to be true that one can write the schroedinger equation in this notation, although it might be clunky.
But okay, here’s an explicit example. I’m going to stick with discrete examples because this ket-like notation is much more suited to those, but I’ll work in the position basis in accordance with your wishes.
So: we have two wells a particle can sit in, (a) and (b). It can also tunnel between them, so energy eigenstates are the symmetric and antisymmetric combinations of the wells: (E1)=50%(A)+50%(B,pi), and (E2)=50%(A)+50%(B,0).
The Schrodinger equation says: d/dt (E)=(E, E*t). I’ve set hbar=1 for convenience. Let’s apply to this our system by assuming that a particle starts in (a) and asking what state it is in after time t.
(I’m using the prescribed rules for how the angle algebra should go, which I haven’t shown you explicitly but is hopefully clear). So after time t, we have:
And there you have it, time evolution via Schroedinger, represented as probabilities the whole way through. As before, the compromise we have to make is shifting the stuff that actually matters into these phases that are not especially transparent.
Nonetheless, they do have a consistent set of rules. If you’re willing to accept those as just the way that branches must be combined, handed down from above, then you can track the shifting of these branches without any additional difficulty. As I said before, I leave it to your judgement whether this is a meaningful way of avoiding amplitudes to you or not.
It's a fine way of avoiding amplitudes, if that floats your boat, but it just doesn't get at the question I'm asking. As you say, you have to accept the way the branches combine, which is no longer linear, and isn't derived from amplitude branch counting.
