1D TDSE Wavepacket Scattering

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 $\hbar = m = 1$ .

Figure 1. Gaussian wavepacket scattering off a rectangular barrier under the 1D TDSE. Method: Crank-Nicolson on 800-point uniform grid, tridiagonal Thomas.

potential

V_04.0

k_02.0

speed6

WHAT TO TRY

Lower k_0 so the incident energy E = k_0 squared over 2 drops below the barrier height V_0: the packet almost entirely reflects and only a faint tunnelling ripple leaks through. The T(E) diagnostic puts the operating point deep in the forbidden region where transmission is exponentially small.
Raise k_0 past E = V_0: transmission jumps up and then oscillates with energy. Those bumps are resonances where the barrier width is a half-integer number of wavelengths, and the operating dot rides the analytic T(E) curve through them.
Switch the potential to the square well: now the Ramsauer-Townsend resonances appear, energies where the well turns perfectly transparent (T = 1). Each setting drops the simulated transmission point right onto the analytic curve.