Quantum Tunneling & Real-World Applications
Chapter 6 established that a quantum particle simply doesn't possess a sharply defined position and momentum at once. This chapter puts that same underlying wave nature to work explaining something even stranger: a particle genuinely passing through a barrier that classical physics says it has no possible way to cross.
Classically Forbidden, Quantum Mechanically Real
In classical physics, a ball rolling toward a hill it doesn't have enough energy to climb simply rolls back down — it cannot appear on the far side. A quantum particle facing an equivalent energy barrier behaves differently. Because its own wave function doesn't drop to exactly zero the instant it meets a barrier, but instead decays smoothly within it, there is a real, nonzero probability of the particle appearing on the far side — a phenomenon called quantum tunneling.
The Real 1927–1928 History
Friedrich Hund, in 1927, was the real first physicist to apply the Schrödinger equation to a tunneling-type problem, studying a double-well potential in the context of molecular spectra. The real breakthrough into nuclear physics came the following year: in 1928, George Gamow provided a real, successful explanation of alpha decay using quantum tunneling, while Ronald Gurney and Edward Condon independently solved the Schrödinger equation for the same nuclear problem, deriving a real relationship between a radioactive particle's half-life and the mathematical probability of it tunneling out of the nucleus.
The Transmission Coefficient
The real probability of a particle tunneling through a barrier is captured by its transmission coefficient. For a simple rectangular barrier of height V0 (above the particle's own energy E) and width L:
This real relationship shows that tunneling probability falls off exponentially with barrier width, barrier height, and particle mass — which is exactly why tunneling is a genuinely significant, observable effect for light particles like electrons, and becomes vanishingly small for anything heavier or macroscopic.
Worked Example: Electron Tunneling Through a Thin Barrier
An electron (mass 9.109 × 10-31 kg) approaches a barrier 1 eV (1.602 × 10-19 J) higher than its own energy, with a width of 1 nm (1 × 10-9 m). Estimate the transmission probability.
κ ≈ 5.12 × 109 m-1
T ≈ e-2(5.12×109)(1×10-9) = e-10.25 ≈ 3.5 × 10-5
Roughly a 1-in-30,000 chance per attempt — small, but real and nonzero, and, crucially, not zero the way classical physics would predict.
Real, Documented Applications
- Alpha radioactive decay — an alpha particle tunnels out through the strong nuclear force's own confining barrier, exactly as Gamow, Gurney, and Condon first explained.
- The scanning tunneling microscope (STM) — invented in 1981 by Gerd Binnig and Heinrich Rohrer, the STM measures the real tunneling current between a sharp conducting tip and a surface, achieving a real resolution of roughly 0.001 nm — fine enough to image individual atoms.
- Tunnel diodes — real semiconductor devices that exploit tunneling current, which drops off rapidly as voltage increases, giving them a genuinely unusual current-voltage behavior used in high-speed electronics.
A Real Callback: Tunneling and Stellar Fusion
Quantum tunneling resolves this real gap directly. Even though any individual proton-proton collision has only a small real chance of tunneling through the Coulomb barrier, the Sun's core contains an astronomically large number of protons colliding constantly — and that same large number, multiplied against a small but genuinely nonzero per-collision tunneling probability, is enough to sustain the Sun's own real, steady fusion rate. Without quantum tunneling, stars like the Sun simply would not shine.
Classical vs. Quantum Barrier-Crossing
| Property | Classical Physics | Quantum Tunneling |
|---|---|---|
| Crossing a barrier taller than the particle's energy | Impossible | Possible, with a real, calculable probability |
| Depends on | Only whether energy exceeds the barrier | Barrier height, width, and particle mass (exponentially) |
| Real-world consequence | The Sun's core would be too cool to fuse hydrogen | Stellar fusion proceeds despite the Coulomb barrier |
Hands-On Exercises
Quick Reference
- Quantum tunneling: a real, nonzero probability of a particle crossing a classically forbidden barrier
- Real history: Hund (1927, double-well potentials), Gamow, Gurney & Condon (1928, alpha decay); the term "tunnel effect" dates to Frenkel's 1932 textbook
- Transmission probability falls off exponentially with barrier height, width, and particle mass
- Real applications: alpha decay, the scanning tunneling microscope (1981), tunnel diodes
- Real callback: quantum tunneling is what allows the Sun's core to sustain fusion despite classically insufficient thermal energy (Astronomy Fundamentals Chapter 4)