Not who you asked (and I don't quite understand everything) but I think that's about right, except in the continuous world. You pick an encoding scheme (either the Lagrangian or the Hamiltonian) to go from state -> vector. You have a "rules" matrix, very roughly similar to a Markov matrix, H, and (stretching the limit of my knowledge here) exp(-iHt) very roughly "translates" from the discrete stepwise world to the continuous world. I'm sure that last part made more knowledgeable people cringe, but it's roughly in the right direction. The part I don't understand at all is the -i factor: exp(-it) just circles back on itself after t=2pi, so it feels like exp(-iHt) should be a periodic function?
Yes, exp(-iHt) means the vector state is rotating as time passes, and it rotates faster when the Hamiltonian (energy) is bigger. This rotation gives the wave like behavior.
Slightly related, there is an old video of Feynman where he tries to teach quantum mechanics to some art students, and he explains this complex rotation and its effects without any reference to math.
