Particles pass through barriers they classically cannot surmount—the wave function doesn't care about walls
Imagine rolling a ball toward a hill. If the ball doesn't have enough energy to reach the top, it rolls back—always. There's no "small chance" it appears on the other side. The energy constraint is absolute.
The Schrödinger equation describes how quantum states evolve. For a particle approaching a barrier higher than its energy, the solution in the barrier region is not oscillatory but exponentially decaying:
The key insight: this decay doesn't mean "the particle stops." The wave function has continuous value and slope at the barrier boundaries. If the barrier is narrow, the decaying exponential hasn't reached zero before exiting—and an oscillatory solution continues on the far side.
The probability of tunneling depends exponentially on the barrier width and the decay constant κ (which depends on how much the barrier exceeds the particle's energy):
A slight increase in barrier width causes dramatic decrease in transmission. This exponential sensitivity is crucial: tunneling is significant only for atomic-scale barriers or light particles like electrons.
Radioactive decay: Alpha decay is pure tunneling. The alpha particle bounces around inside the nucleus billions of times per second, each time having a tiny probability of tunneling out. Decay half-lives range from nanoseconds to billions of years—all determined by barrier geometry.
Nuclear fusion in stars: The Sun fuses hydrogen despite temperatures far below what classical physics requires. Protons tunnel through their mutual Coulomb repulsion. Without tunneling, stars wouldn't shine.
Scanning Tunneling Microscope (STM): A sharp metal tip is brought within nanometers of a surface. Electrons tunnel between tip and surface, creating a current exquisitely sensitive to distance. This allows imaging individual atoms—Nobel Prize 1986.
Flash memory: Your USB drive stores data using quantum tunneling. Electrons are forced to tunnel through thin oxide barriers onto floating gates, where they're trapped for years, representing stored bits.
How long does tunneling take? This question has generated decades of debate. Experiments suggest tunneling can occur in less time than it would take light to cross the barrier—but this doesn't violate relativity. The particle's wave function was already present on both sides; the "tunneling time" doesn't correspond to a particle moving through space.
Could you tunnel through a wall? Technically, there's a non-zero probability. Practically, for a human-sized object and a meter-thick wall, that probability is roughly 10−10^30—a number so small it's meaningless. You'd wait longer than the age of the universe raised to the power of the age of the universe. Quantum tunneling is strictly a nanoscale phenomenon.