Sandbox · Fig. 02
1D Schrödinger
Bound states diagonalises Ĥ on a grid and stacks the first six eigenstates at their energies. Scattering launches a Gaussian packet and watches it hit a barrier — partial transmission below V₀ is tunneling in action.
The solid line is the potential V(x); dashed lines are energy levels; the curves drawn on each level show ψn(x) (or its density, depending on the toggle). The harmonic preset should give evenly-spaced levels at (n + ½) ω. The double well gives nearly-degenerate pairs whose splitting is the tunneling rate between the two minima.
T(E) = – · R(E) = – · live L/R probabilities update as the packet propagates.
Quantum mechanics smooths the classical "all or nothing" rule. Below V₀, T(E) is not zero — that exponential leak is tunneling, the same physics that makes alpha decay and STMs work. The double-barrier preset reveals Fabry-Pérot-like resonances at certain energies, where T → 1 even though the barriers are still there — a hint at how quantum-well devices (RTDs) operate.