1D TDSE: wavepacket scattering off a barrier

What you are seeing: a Gaussian quantum wavepacket moving to the right and hitting a potential barrier. Part of the wavefunction reflects, part tunnels through. The standing Schrodinger equation only gives you transmission probabilities at fixed energies (the green theory curve); this simulation watches the full time evolution and shows what really happens to a finite-width pulse: it stretches, interferes with itself, and splits cleanly into reflected and transmitted parts.

We solve $i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m}\partial_x^2\psi + V(x)\psi$ via Crank-Nicolson: $(I + i\,dt\,H/2)\psi^{n+1} = (I - i\,dt\,H/2)\psi^n$. This is unconditionally stable and norm-preserving to 1e-10 per step. Blue curve: $\mathrm{Re}\,\psi$. Red curve: $|\psi|^2$. Grey rectangle: barrier. Code units $\hbar = m = 1$.