Quantum tunnelling

Can a quantum particle pass through a barrier taller than its own energy — a wall it classically could never cross?

▶ Run the simulationSee the measured result

Units: transmission probability T at E = 2.88, V₀ = 3.2, a = 1.6 (κ = 0.8 exactly, κa = 1.28)

How the lab tests it

Launch a Gaussian wavepacket at a potential barrier with energy E < V₀ and evolve the Schrödinger equation (Visscher's norm-conserving scheme); measure the probability T that ends up beyond the barrier, and compare to the rectangular-barrier formula.

What it checks

T > 0 (classically T = 0 for E < V₀ — the particle always bounces) matching the tunnelling coefficient, while the total probability ∫|ψ|² stays 1 (unitary evolution)

Quantum tunnelling calculator: transmission coefficient, decay length & barrier width

A wall taller than the particle, and the particle comes out the other side. That is the whole of this page, and every number on it is built from the barrier rather than stored: the energy is assembled the way the simulation above assembles it, E = k₀²/2, and the transmission follows from κ = √(2m(V₀−E))/ħ and nothing else. Read one way it answers the question a course sets — what fraction gets through a rectangular barrier of height V₀ and width a — and read the other way it answers the question an instrument asks, which is the useful one: a current came back, how wide is the gap? That inverse is a scanning tunnelling microscope, and the same exponent that makes T impossible to guess makes a easy to measure to the picometre. In between sit the two things tunnelling is famous for. Gamow's wall, where lnT falls linearly in the width at a slope of −2κ, is why alpha-decay half-lives run from microseconds to 10¹⁷ years while the energies that set them barely double. And the Ramsauer–Townsend resonance, where a barrier the wave has more than enough energy to cross turns COMPLETELY transparent whenever a whole number of half-waves fits inside it — exactly one, in the arithmetic as well as in the physics, and this page shows why that is not a rounding. Planck's constant is a dial here, not a decoration: at ħ = 1 every κ is the shipped expression divided by one, and at ħ = 0 it is infinite, sinh is infinite and T is EXACTLY zero — Newton's particle, reached by turning the constant off rather than by writing a second formula, and falsified on both sides of the barrier. The measured boxes hold the lab's own recovered numbers, and one predicate decides whether they are quoted at all: move k₀, V₀, a or ħ off the configuration the oracle pinned and the page withdraws the comparison instead of printing a meaningless one. Four things it will not do: barriers of any other shape (a real nucleus is a Coulomb tail, which is an integral rather than a width); resonant tunnelling through two barriers, where the transparency is a bound state rather than a half-wave count; the Coulomb blockade and everything else about tunnelling that is about charge rather than about waves; and any claim about the simulation's own reading beyond the one the last direction re-executes.