An electron wave packet (a Gaussian of plane waves) runs into a potential. The time-dependent Schroedinger equation is solved on a 60 nm grid; the bright curve is the probability density, drawn at the packet's central energy, the grey shape is the potential. Part of the packet tunnels through even though its energy is below the top. Units: nm, eV, fs; free-electron mass.
Fires a packet at 9 mean energies, measures the transmitted probability after the packet has passed, and compares it with the transfer-matrix transmission averaged over the packet's momentum spread.
Press the button.
The science
The equation. iℏ ∂ψ/∂t = -(ℏ2/2m) ∂2ψ/∂x2 + V(x) ψ with ℏ2/2m = 0.0381 eV nm2 for a free electron. The packet ψ ∝ exp(-(x-x0)2/4σ2) ei k0 x has central energy E = ℏ2k02/2m (the slider; the true central energy is higher by ℏ2/8mσ2, checked numerically) and momentum spread 1/(2σ).
Solver. Strang split-step Fourier method: half a potential step, a full kinetic step in momentum space (exact), half a potential step; 1024 points, time step 0.04 ℏ/eV (0.026 fs). It conserves norm and energy to round-off in free space (checked). Absorbing layers 4 nm thick at both edges remove outgoing waves; potential edges are smoothed over one grid cell (0.06 nm).
Tunnelling. For a rectangular barrier with E < V0, T = [1 + V02 sinh2(κ a) / (4E(V0-E))]-1 with κ = √(2m(V0-E))/ℏ; for thick barriers T ≈ e-2κa (the page checks the slope of ln T against width). For E > V0 the sinh becomes sin: reflection appears and T = 1 exactly when k2a = nπ.
Transfer matrix. For any piecewise-constant potential (the Gaussian is cut into 80 slices) the transmission is found by matching ψ and ψ' at every interface. This is a separate calculation from the time-dependent solver, and the sweep shows that a packet's transmitted probability equals the matrix result averaged over its momentum distribution (∫ |φ(k)|2 T(k) dk).
Resonant tunnelling. Two barriers enclose a gap that holds quasi-bound states; when the packet energy matches one, waves bounce between the barriers in phase and the transmission rises to 1 even though one barrier alone transmits only a few percent. The preset finds the first resonance numerically. This is the principle of the resonant-tunnelling diode.
What is simplified. One dimension, a free-electron mass, ideal rectangular or Gaussian barriers (no effective mass, band structure or interactions). A packet spread over a range of energies means measured T differs from the single-energy T(E) near sharp resonances (the page shows both). Transmitted probability is read after the packet has passed; for a double barrier a little probability can still sit between the barriers, so T + R can be slightly below 1. The screen shows probability density only; phase is available with "Show Re(psi)". Not a device simulator.
Canvas 2D simulation, no 3D assets. Educational model.