Quantum Tunneling: How Particles Pass Through Walls

An alpha particle sits in the nucleus of uranium-238, trapped inside a spherical potential well created by the strong nuclear force. To escape, it would need to climb an energy barrier taller than its own kinetic energy. Classical physics says it stays trapped forever. Yet every 4.5 billion years, the uranium nucleus emits an alpha particle anyway. The particle does not break the well; it simply vanishes from inside and appears outside, having somehow passed through the barrier. This is quantum tunneling: one of the most counterintuitive predictions of quantum mechanics, and one of the most experimentally confirmed. It is the mechanism behind radioactive decay, nuclear fusion in stars, and the billions of transistors in every modern computer chip.

What Classical Physics Predicts

A particle confined in a potential well has kinetic energy. To escape the well and reach the outside world, it must have kinetic energy greater than the height of the barrier—the difference in potential energy between the inside and the outside.

Potential energy — the energy a particle has due to its position in a force field, such as the electrostatic repulsion between two like charges.

In classical mechanics, this is absolute. A ball rolling in a valley cannot reach a hilltop higher than its kinetic energy allows. An electron in a metal cannot jump across a vacuum gap unless it has enough energy. A nucleus cannot emit an alpha particle if the alpha particle does not have enough energy to escape the nuclear binding force. The classical prediction is unambiguous: no tunneling.

Yet in 1928, physicist George Gamow applied quantum mechanics to alpha decay and calculated a nonzero probability that an alpha particle could be found outside a nucleus even when its kinetic energy was too low to escape classically. The prediction was startling. Within a few years, experiments confirmed it. Particles do tunnel.

Source: Gamow, G. (1928), "Zur Quantentheorie des Atomkernes," Zeitschrift für Physik, 52(7–8), 510–515 — the foundational paper deriving the tunneling rate for alpha decay from the Schrödinger equation.

How Wave Functions Penetrate Barriers

Quantum mechanics describes a particle not as a point with a definite position, but as a wave function—a mathematical object that assigns a probability to finding the particle at each location. The wave function must satisfy the Schrödinger equation, which relates how the wave function varies in space and time to the potential energy at that location.

Wave function — in quantum mechanics, a mathematical description of the state of a particle that encodes the probability of finding the particle at any position. Denoted ψ(x), its square |ψ(x)|² gives the probability density.

When a particle encounters a potential barrier, the wave function does not simply stop at the barrier's edge. Instead, it continues across the barrier, decaying exponentially as it penetrates deeper into the classically forbidden region. On the far side of the barrier, the wave function is much smaller—but not zero. The particle, being described by this wave function, has a nonzero probability of being found on the far side.

Think of the wave function like the ripples on a pond hitting a partial barrier. The wave does not reflect completely; part of it leaks through, weaker on the far side but still present. A ball rolling toward a hill does not work this way—it either has enough energy to climb the hill or it does not. A probability wave is different. Its amplitude does not vanish at the barrier; it decays.

The penetration distance into the classically forbidden region is set by the de Broglie wavelength—shorter wavelengths penetrate less deeply. A heavier particle, at the same energy, has a shorter wavelength and penetrates less. A lighter particle has a longer wavelength and penetrates farther.

Source: Griffiths, D. J. (2005), Introduction to Quantum Mechanics, 2nd ed., Chapter 2 — detailed derivation of tunneling probability from the time-independent Schrödinger equation.

Gamow's Calculation: Alpha Decay

In 1928, George Gamow (then 24 years old) realized that the Schrödinger equation, applied to an alpha particle confined in a nucleus, predicted nonzero probability of finding the alpha outside the nucleus—even at energies where classical mechanics forbids it.

The calculation is elegant. An alpha particle in a nucleus experiences two forces: the strong nuclear force (attractive, short-range, keeping it bound) and the Coulomb repulsion (the electrostatic repulsion between the positively charged alpha and the positively charged nucleus). The result is an effective potential well—a deep attractive region near the nucleus's center, surrounded by a Coulomb barrier.

When an alpha particle has kinetic energy lower than the Coulomb barrier height, it is trapped. Classically, it bounces around inside the nucleus forever, unable to escape. Quantum mechanically, the wave function leaks through the barrier. Gamow calculated the leak rate—the tunneling probability per unit time—and found it agreed with the observed lifetimes of alpha-emitting nuclei.

Coulomb barrier — the electrostatic repulsion between two like-charged particles, which creates a potential energy barrier. For nuclei, this barrier opposes the strong force that would otherwise bind protons together.

For uranium-238, the alpha particle has a kinetic energy of about 4.3 MeV, but the Coulomb barrier is about 6 MeV high—the particle is classically forbidden to escape. Yet the decay lifetime is 4.5 billion years. The decay happens via tunneling.

Gamow's calculation showed that the tunneling probability depends exponentially on a factor involving the barrier height, width, and the alpha particle's mass and energy:

P ≈ exp(−2πη)

where η is a dimensionless parameter depending on the nuclear charge, the alpha particle energy, and fundamental constants. Small changes in η produce enormous changes in P—the exponential dependence is very steep.

Source: Gamow, G. (1928), Zeitschrift für Physik, 52(7–8), 510–515 — derivation of tunneling probability including the exponential factor.

Source: Krane, K. S. (1988), Introductory Nuclear Physics, Wiley, Chapter 6 — modern treatment of alpha decay and tunneling in nuclei.

Why It Depends Exponentially on Everything

The exponential dependence is the crucial feature of quantum tunneling. It means that small changes in any parameter produce enormous changes in the tunneling rate.

If the barrier is slightly higher, the probability drops exponentially. If the particle is slightly heavier, the probability drops exponentially. If the barrier is slightly wider, the probability drops exponentially. This is why tunneling is significant at subatomic scales (where barriers are thin and particles are light) but vanishes for macroscopic objects (where barriers are thick or particles are heavy).

A tennis ball, classically confined in a box, has a de Broglie wavelength far smaller than an atomic nucleus. The barrier width is macroscopic (the wall thickness), and the penetration distance is negligible. The tunneling probability is incomprehensibly small—far, far smaller than 10^−100. Quantum mechanics predicts a nonzero probability, but it is so small that a quantum computer given the age of the universe could not generate a single such event by random chance.

An electron crossing a thin barrier (perhaps 1 nanometer wide) in a modern transistor has a much longer de Broglie wavelength and a much smaller mass. The tunneling probability is often tens of percent. This is why quantum tunneling is practically important in electronics.

This exponential sensitivity to parameters is why quantum tunneling is such an elegant fingerprint of quantum behavior. In classical physics, the tunneling probability would be either zero or one—particle crosses or does not. In quantum mechanics, the probability is a function of a few parameters, and its exponential form is distinctive.

Scenario	Barrier Width	Particle Mass	Approximate Tunneling Probability
Tennis ball through wall	meters	kg	10^(−10^30) (impossibly small)
Electron through 1 nm barrier	1 nm	9.1 × 10^(−31) kg	10% to 50%
Alpha particle escaping uranium nucleus	femtometers	6,600 × electron mass	10^(−22) per second (4.5 billion year half-life)

Direct Experimental Confirmation

Within a few years of Gamow's 1928 paper, experiments confirmed quantum tunneling. The first direct test came from observations of alpha decay itself. Radioactive nuclei emit alpha particles with kinetic energies lower than the Coulomb barrier height—exactly as tunneling theory predicted and classical physics forbade.

More convincingly, physicists measured the half-lives of different alpha-emitting isotopes and compared them to the tunneling probability formula. The agreement was striking. Nuclei with slightly lower alpha particle energies (which should tunnel less) were observed to have much longer lifetimes. The relationship matched the exponential dependence predicted by tunneling theory with remarkable precision.

Source: Condon, E. U. & Shortley, G. H. (1935), The Theory of Atomic Spectra, Cambridge University Press, Chapter 18 — early experimental confirmation of alpha decay via tunneling theory.

Tunneling in Nuclear Fusion

The sun's energy comes from nuclear fusion: the collision and merger of hydrogen nuclei (protons) to form deuterium and heavier nuclei. This process releases enormous energy, powering stellar interiors and eventually enabling life on Earth.

Yet protons, approaching each other, experience strong Coulomb repulsion. The temperature in the sun's core (about 14 million Kelvin) provides particles with kinetic energies of roughly 1 keV each. The Coulomb barrier for two colliding protons is about 400 keV. By classical physics, the protons should almost never fuse. The barrier is too high; the particles lack sufficient energy to climb it and approach closely enough for the strong nuclear force to bind them.

Strong nuclear force — one of the four fundamental forces, responsible for binding protons and neutrons in atomic nuclei. It is attractive, short-ranged (extends only about a femtometer), and overwhelms electrostatic repulsion at very short distances.

Yet fusion happens at a rate sufficient to power the sun. Quantum tunneling is the reason. Protons tunnel through the Coulomb barrier, occasionally finding themselves at distances short enough that the strong force takes over and binds them. Without tunneling, the sun would have burned out billions of years ago—and Earth's life would not exist.

The fusion rate depends exponentially on the barrier height and the particle's energy. A small increase in core temperature dramatically increases the fusion rate. This is why stars reach a stable equilibrium: as a star contracts under gravity, its core heats up, fusion accelerates, outward pressure increases, and equilibrium is restored. The exponential tunneling dependence is what regulates stellar burning.

Source: Kippenhahn, R., Weigert, A., & Weiss, A. (2012), Stellar Structure and Evolution, Springer, Chapter 18 — detailed treatment of tunneling and fusion rates in stars.

Tunneling in Semiconductor Devices

Modern electronics depend on quantum tunneling. The transistor, the most numerous manufactured object in human history (tens of sextillions exist in operation), works by controlling the flow of electrons across potential barriers—barriers that electrons tunnel through.

In a metal-oxide-semiconductor field-effect transistor (MOSFET), electrons tunnel through a thin insulating oxide layer separating the gate and the channel. By applying a voltage to the gate, the device modulates this tunneling current, turning it on and off. Without quantum tunneling, such devices would not work.

The tunnel rate, depending exponentially on barrier thickness, sets strict limits on miniaturization. As transistors shrink, the insulating layers must shrink proportionally. At some point, the barrier becomes so thin that the tunneling probability becomes unacceptably high—electrons leak through the insulator when they should not, wasting power and corrupting logic. This problem, called the "leakage current problem," is one of the central challenges in continuing Moore's Law—the historical trend of doubling transistor density every two years.

Source: International Roadmap for Devices and Systems (IRDS), "Tunnel FET and Beyond CMOS Devices" (2024) — current challenges in managing tunneling leakage in scaled transistors.

Scanning Tunneling Microscope

In 1981, Gerd Binnig and Calvin Quate invented the scanning tunneling microscope (STM), which uses quantum tunneling to image individual atoms. The STM brings a sharp metal needle to within about a nanometer of a conducting surface. When a voltage is applied between the needle and the surface, electrons tunnel through the vacuum gap. The tunneling current is exponentially sensitive to the gap width.

As the needle scans across the surface, it is feedback-controlled to maintain constant tunneling current. This means the needle follows the atomic topography of the surface—raising slightly over an atom, descending slightly into a valley. The needle's height at each point directly maps the surface's atomic structure.

The tunneling current depends exponentially on gap width. A 0.1 nm change in gap width can change the current by an order of magnitude. This sensitivity is what gives the STM its remarkable spatial resolution: better than 0.1 nanometers, allowing imaging of individual atoms. No optical microscope can achieve this resolution, because light wavelengths are much larger than atoms.

Binnig and Quate shared the 1986 Nobel Prize in Physics for the STM. The device transformed nanoscience, enabling researchers to manipulate individual atoms and molecules and to visualize atomic-scale structures that had previously been invisible.

Source: Binnig, G., Quate, C. F., & Gerhard, D. P. (1986), "Atomic Force Microscope," Physical Review Letters, 56(9), 930–933 — the original STM paper.

The Quantum Eraser and Tunneling

A remarkable experiment, mentioned in passing in the broader context of quantum measurement, involves the quantum eraser—a setup where you can erase which-path information after tunneling events have occurred. In such experiments with tunneling systems, erasing which-path information has been observed to restore interference patterns that were destroyed by the measurement of the tunnel pathway.

See The Quantum Eraser Experiment for details on how measurement and erasure affect quantum behavior.

Tunneling and Radioactive Dating

Radioactive decay via tunneling provides the basis for radiocarbon dating and other radiometric age-measurement techniques. Carbon-14, produced in the upper atmosphere by cosmic rays, decays by beta decay (a form of tunneling involving weak nuclear force interactions) with a half-life of 5,730 years. By measuring the ratio of carbon-14 to carbon-12 in organic remains, archaeologists and paleontologists determine when the organism died.

The constant half-life of carbon-14—arising from the constant tunneling probability per nucleus per unit time—provides a reliable clock. Without understanding tunneling, radiometric dating would be a mystery. With tunneling theory, the decay rate and half-life follow directly from quantum mechanics and the properties of the nucleus and the emitted particle.

Source: Libby, W. F. (1955), Radiocarbon Dating, University of Chicago Press — foundational work connecting radioactive decay rates to tunneling probability.

What Tunneling Does Not Explain

Tunneling is sometimes misrepresented. It does not allow macroscopic objects to vanish and reappear. A person sitting in a chair has zero probability of tunneling through the floor, not because tunneling is forbidden but because the exponential factor is so small that it is unphysical. Quantum mechanics does not violate classical physics at macroscopic scales; rather, quantum predictions approach classical ones as objects become larger.

Tunneling also does not violate energy conservation. A particle tunneling through a barrier does not gain kinetic energy. In quantum mechanics, the energy of a particle is its total energy—the sum of kinetic and potential energy—and this is conserved on average. In a tunneling event, the particle temporarily has negative kinetic energy (velocity imaginary) in the classically forbidden region, but only during the brief quantum tunneling process.

Tunneling probability depends sensitively on barrier parameters. Two nuclei differing by only a few percent in barrier height can differ by orders of magnitude in half-life. This exponential sensitivity is often surprising to those first encountering tunneling, but it is a direct consequence of the wave function's exponential decay in the classically forbidden region.

The Takeaway

Quantum tunneling is a quantum mechanical effect with no classical analog. It allows particles to pass through barriers that classical physics forbids, with a probability determined by the barrier height, width, and the particle's mass. The tunneling probability depends exponentially on these parameters, making it rare for macroscopic objects but frequent at atomic scales. Tunneling explains radioactive decay, powers fusion in stars, enables semiconductor electronics, and revealed atomic structure through the scanning tunneling microscope. It is a counterintuitive but thoroughly tested prediction of quantum mechanics, and one of its most practically important implications.

Frequently Asked Questions

Q: Can macroscopic objects like a person tunnel through a wall?
A: Quantum mechanics allows it with nonzero probability, but the probability is unimaginably small—far smaller than 10^−100 for any realistic human-wall system. The exponential dependence of tunneling on particle mass means that large, heavy objects have negligible tunneling probability. You will not observe a person tunneling through a wall, not because the laws of physics forbid it, but because the probability is smaller than any physically meaningful timescale.

Q: Does tunneling violate energy conservation?
A: No. In quantum mechanics, a particle's energy is its total energy (kinetic plus potential). During tunneling through a classically forbidden region, the particle's kinetic energy is temporarily negative (the velocity becomes imaginary in quantum mechanical terms), but total energy is conserved on average. Tunneling is consistent with energy conservation when understood in the quantum mechanical framework.

Q: How does temperature affect tunneling?
A: Higher temperature increases the kinetic energy of particles approaching a barrier. With higher kinetic energy, the tunneling probability increases (the exponential factor decreases). This is why fusion rates in stars increase dramatically with temperature—even a small temperature increase produces a large exponential increase in tunneling probability. This temperature sensitivity is essential for stellar stability.

Q: What is the difference between tunneling and classically allowed particle motion?
A: A classically allowed particle, with kinetic energy greater than the barrier height, passes over the barrier. Its probability of being found on the far side is essentially one hundred percent. A particle tunneling through a barrier has kinetic energy less than the barrier height, yet has a nonzero (but typically much smaller) probability of being found on the far side. The probability depends exponentially on barrier parameters.

Q: Can tunneling be observed directly?
A: Alpha decay is direct evidence: alpha particles escape from nuclei with kinetic energies lower than the Coulomb barrier height, a phenomenon that classical physics forbids and tunneling theory predicts. Scanning tunneling microscopes directly use tunneling to achieve atomic-scale imaging. Semiconductor devices rely on controlled tunneling. All of these provide direct experimental evidence for tunneling.

Q: Why is tunneling important in modern electronics?
A: Transistors use tunneling to control current flow across potential barriers. As transistors shrink, insulating layers become thinner and tunneling current increases. Managing this leakage current is one of the major engineering challenges in continuing to miniaturize chips. Understanding and controlling tunneling is essential for further advances in semiconductor technology.

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