Quantum Tunneller

Quantum Tunnelling — The Complete Physics Guide

In classical mechanics, a ball rolling toward a hill it doesn't have enough energy to climb simply rolls back down — it can never appear on the other side. Quantum mechanics breaks this rule completely. A particle with energy less than a barrier's height still has a genuine, calculable probability of appearing on the far side, a phenomenon called quantum tunnelling. Quantum Tunneller asks you to work with the exact formula that predicts this probability.

Why Tunnelling Happens

A quantum particle is described by a wavefunction, and the square of that wavefunction gives the probability of finding the particle at any location. Outside a barrier, where E > V(x), the wavefunction oscillates like a free wave. But inside a barrier taller than the particle's energy, the wavefunction doesn't drop to zero — instead it decays exponentially, following e^(−κx) where κ = √(2(V₀−E)) in natural units. As long as the barrier isn't infinitely wide, some of that decaying wavefunction survives to reach the far side, where it becomes an oscillating wave again — just with much smaller amplitude.

The transmission probability, in the standard thick-barrier approximation, is T = 16(E/V₀)(1−E/V₀)e^(−2κL). The prefactor 16(E/V₀)(1−E/V₀) is largest when E = V₀/2, but the exponential term e^(−2κL) usually dominates the overall behavior — meaning both higher energy and thinner barriers matter enormously.

The phenomenon was first worked out theoretically in 1928, when George Gamow — and independently Ronald Gurney and Edward Condon — used exactly this barrier-penetration mathematics to explain alpha decay, one of the earliest triumphs of applying the new quantum mechanics to a real nuclear physics problem. Before that explanation, the enormous range of observed half-lives across different radioactive isotopes (from microseconds to billions of years) had no satisfying classical explanation; the exponential sensitivity of T to barrier width and height turned out to explain that entire range naturally, since even small differences in the effective nuclear barrier translate into astronomically different tunnelling probabilities.

Why Width Matters So Much More Than It Seems

Because T depends on e^(−2κL), the relationship between barrier width and tunnelling probability isn't linear or even polynomial — it's exponential. Doubling the barrier width doesn't halve the probability; it can reduce it by many orders of magnitude, depending on κ. This exponential sensitivity is exactly what makes scanning tunnelling microscopes work: moving a probe tip by a single nanometer above a surface changes the tunnelling current dramatically, giving atomic-scale resolution from a purely quantum effect.

This same exponential sensitivity explains why tunnelling is completely unobservable at everyday scales. For a macroscopic object like a ball or a person, the effective κL for any real barrier is so enormous that the transmission probability rounds to zero at any precision that could ever be measured — not because the physics stops applying, but because the exponential suppression becomes total.

What the Wavefunction Actually Represents

It's worth being precise about what is and isn't happening physically during tunnelling, because the popular description — "the particle passes through the wall" — can be misleading. Quantum mechanics doesn't describe a particle as a tiny billiard ball with a definite trajectory; it describes the particle by a wavefunction, and the square of that wavefunction's magnitude gives the probability of finding the particle at any given location if you were to measure it there. Inside the barrier, the wavefunction is a real, mathematically well-defined evanescent wave — it simply corresponds to a rapidly decreasing probability of detection, not to the particle physically "burrowing" through solid material the way a mole tunnels through soil.

This distinction matters because it resolves an apparent paradox: classically, a particle inside the barrier region would need negative kinetic energy (since its total energy E is less than the potential V₀ there), which sounds physically impossible. Quantum mechanically, there's no contradiction, because the particle is never actually measured to be "inside the barrier with negative kinetic energy" — the wavefunction inside the barrier is a mathematical amplitude, not a description of the particle's classical trajectory at each instant. Only the amplitude that survives to the far side of the barrier corresponds to something an experiment could actually detect.

A Common Misconception: Tunnelling Is Not "Borrowing" Energy

A frequently repeated but inaccurate explanation of tunnelling invokes the energy-time uncertainty principle, claiming the particle "borrows" enough energy to briefly climb over the barrier classically, then "pays it back." This picture is popular because it sounds intuitive, but it isn't how the standard quantum mechanical treatment actually works, and it isn't necessary to explain the effect. The Schrödinger equation calculation used in this game never requires the particle's energy to exceed E at any point — the decaying wavefunction inside the barrier is an exact, energy-conserving solution, not a temporary violation of energy conservation.

The correct picture is simpler, if less viscerally satisfying: the particle's energy stays exactly E throughout, both outside and (in a formal sense) inside the barrier. What changes inside the barrier is that the wavefunction's character shifts from oscillatory to exponentially decaying, which is precisely what the mathematics of the Schrödinger equation predicts whenever a particle's total energy is less than the local potential energy — no energy borrowing, no classical excursion above the barrier, just the ordinary rules of wave mechanics applied consistently.

Worked Example — Computing Tunnelling Probability

Problem: A particle with energy E = 9.33 (natural units) approaches a barrier of height V₀ = 10 and width L = 0.8. What is the tunnelling probability?

κ = √(2(10 − 9.33)) = √1.34 ≈ 1.158

T = 16 × (9.33/10) × (1 − 9.33/10) × e^(−2 × 1.158 × 0.8) ≈ 0.157, or about 15.7%

Real-World Applications

Alpha decay: An alpha particle inside a heavy nucleus is classically trapped by the strong nuclear force, but it can quantum-tunnel through the surrounding Coulomb barrier — the same mathematics used in this game explains why some isotopes decay in microseconds and others take billions of years, based entirely on tiny differences in effective barrier height and width.

Nuclear fusion in stars: Protons in a star's core rarely have enough classical kinetic energy to overcome their mutual electrostatic repulsion, but quantum tunnelling gives them a real (if small) chance of fusing anyway — without tunnelling, stars like the Sun would not shine at anywhere near their observed rate.

Flash memory and tunnel diodes: Modern flash storage stores bits by controlling electron tunnelling through a thin insulating oxide layer — engineers deliberately choose the layer's thickness so that tunnelling is rare enough to hold a charge for years, but controllable enough to erase and rewrite on demand.

Scanning tunnelling microscopy: Invented in 1981, the STM images individual atoms on a conducting surface by holding a sharp metal tip a fraction of a nanometer above the sample and measuring the tunnelling current that flows across the gap — a direct, practical readout of exactly the T = 16(E/V₀)(1−E/V₀)e^(−2κL) relationship this game teaches, with the barrier being the vacuum gap itself.

Frequently Asked Questions