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.
WHAT TO TRY
- Watch the barrier split the packet: part reflects into a left lobe and part tunnels through to the right, with a little trapped briefly inside. The rainbow phase stripes wind fastest where the packet moves fastest.
- Switch the potential: a free particle just spreads, a harmonic well makes a coherent state slosh back and forth, a double well lets the packet tunnel between the two minima. Same Crank-Nicolson solver, a different quantum story each time.
- Raise the barrier V0 above the packet energy E = k0 squared over 2: transmission collapses to a thin tunnelling tail and the T readout drops. Lower it below E and most of the packet sails through.