Quantum Tunneling Time — the Hartman Effect

Barrier & Particle

Barrier height V₀ 3.0 eV

Particle energy E 1.2 eV

Barrier width L 1.20 nm

Tunneling regime (E < V₀): the wavefunction decays exponentially inside the barrier.

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Transmission T

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Reflection R

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Phase (Büttiker–Landauer) time τ

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Effective traversal speed L/τ

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How it works

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.

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).

Bars show |ψ(x)|² (probability density); the animated hue 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.

|ψ(x)|² probability density barrier region V₀

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