A stationary-state solution of the 1D Schrödinger equation is computed exactly for a rectangular barrier V(x)=V₀ on 0<x<L, matching ψ and ψ′ at both interfaces:
ψ_I = e^(ikx) + r e^(-ikx) x < 0
ψ_II = A e^(qx) + B e^(-qx) 0 ≤ x ≤ L
ψ_III = t e^(ik(x-L)) x > L
k = √(2mE)/ħ, q = √(2m(V₀-E))/ħ (real ⇒ tunneling, imaginary ⇒ over-barrier)
r, A, B, t solve a 4×4 complex linear system from continuity of ψ and ψ′ at x=0 and x=L — no approximation, no perturbation theory. (Verified numerically against the closed-form Griffiths transmission formula T=[1+V₀²sinh²(κL)/(4E(V₀-E))]⁻¹ — matches to machine precision, and T+R=1 exactly, so the linear solve here is unchanged from the reference engine.)
Phase (tunneling) time — the Büttiker–Landauer / Wigner phase-delay time is how long the transmission phase φ(E)=arg(t) takes to build up as energy changes:
τ(L) = ħ · dφ/dE (computed here by central finite difference)
The Hartman effect — for an opaque barrier (qL ≳ 3) tanh(qL)→1 and τ saturates to a constant that no longer grows with L. Since the effective speed is L/τ, it then grows without bound as L increases — appearing to exceed c. This does not transmit information faster than light: τ is a phase-delay between two peaks, not a true signal-front velocity, and the transmitted peak is exponentially attenuated (real transmission experiments confirm the phase-time saturation but not any causality violation).
- The flat plot's curve height is |ψ(x)|² (probability density); its color traces arg(ψ(x)) — a genuine traveling-wave phase, not a real-time clock (τ itself is femtoseconds, far too fast to animate 1:1).
- Push E above V₀ to leave the tunneling regime — the barrier becomes classically transparent and shows Ramsauer–Townsend transmission resonances instead.
- Widen L at fixed E<V₀ and watch τ flatten out while T collapses exponentially — that decoupling is the Hartman effect itself.
- Drag the plot to pan, scroll/pinch to zoom the x-axis.