Quantum Wave Packet (1D Schrödinger)

Solves the 1D time-dependent Schrödinger equation i∂ψ/∂t = −½∂²ψ/∂x² + V(x)ψ (ħ = m = 1) with the split-step Fourier method. A Gaussian wave packet is launched to the right and scatters off the potential you choose.

How to use

• Pick a potential: barrier (tunnelling), well, step, harmonic, or free.
• V₀ sets the barrier/well/step height; width its extent.
• k₀ sets the packet's momentum, so energy E = k₀²/2 (yellow dashed line). Compare E to the barrier height.
• Run and watch reflection and transmission; the readout shows the probability that has passed the barrier (transmission).

Quantum tunnelling: even when E < V₀ a finite part of the packet appears on the far side — impossible classically. Lower/narrower barriers transmit more.

Plot: filled blue curve = probability density |ψ|² (autoscaled); light line = potential V(x); yellow dashed = packet energy. Norm stays ≈1 until the packet reaches the absorbing edges, which remove outgoing probability so it doesn't wrap around the periodic FFT box.

Limitations: educational resolution (512 points), fixed time step, soft absorbing boundaries; transmission is read off after the packet clears the barrier, not from a stationary-state calculation.