TDSE Wavepacket Sculptor

A quantum particle in one dimension is a complex wavefunction $\psi(x,t)$ whose evolution obeys the time-dependent Schrödinger equation (TDSE) $i\hbar \partial_t \psi = -(\hbar^2/2m) \partial_{xx} \psi + V(x) \psi$ (here $\hbar = m = 1$). It is advanced by a norm-preserving (unitary) step, so total probability stays exactly one and the dynamics is genuinely quantum. The main view is the probability cloud $|\psi(x)|^2$ coloured by the local phase $\arg(\psi)$; choosing the potential $V(x)$ shows the textbook behaviours: a free packet spreads and its phase winds, a rectangular barrier splits it into a reflected and a tunnelled part, a harmonic well drives a coherent sloshing state, a double well lets probability tunnel back and forth between the minima, and a periodic lattice produces band-like spreading. A strip below tracks the mean position, illustrating Ehrenfest's theorem.