The Leap Through the Wall: Quantum Tunneling from Gamow's Alpha Decay to the Attosecond Riddle

The Hook: A Ball That Rolls Through the Hill

Picture a ball resting in a valley, trapped between two hills. You give it a push, but not enough to reach the crest. In the world we know, the outcome is unambiguous: the ball rolls partway up the slope, comes to a stop, rolls back, and swings within the valley. It will never appear on the other side. To clear the hill you need exactly as much energy as the summit is high — not one bit less. That is the iron bookkeeping principle of classical physics: whoever doesn't bring enough energy doesn't get through. Full stop.

And now the outrageous part: in the quantum world, the ball sometimes does come out on the other side. It doesn't leap over the hill — it simply appears behind it, as though someone had dug a tunnel through rock that doesn't exist. At no point did the ball have enough energy to reach the top; and yet there it is, on the far side. This process is called quantum tunneling, and it is among the effects in which quantum mechanics breaks most sharply with everyday intuition. No trick, no borrowed energy, no secret climbing over — a thoroughly legal, many-times-measured physical event.

What sounds like a curiosity for physics textbooks is in truth one of the most consequential effects there is. Without tunneling the Sun would not shine, because the protons in its core could never overcome their electrical repulsion. Without tunneling there would be no radioactive clocks by which we date rocks and the Earth itself. Without tunneling no one would ever have imaged a single atom on a surface, and the USB stick in your pocket could not hold its data. Tunneling is not the exotic fringe of physics — it is a load-bearing pillar of matter, of technology, and, as recent research shows, even of life.

This article takes you along the whole route: from the question of why a particle can pass through a wall at all, through George Gamow's brilliant explanation of alpha decay in 1928, the furnace of the stars, and the quiet revolution in semiconductor technology, to a question that still provokes dispute today and became tractable only with attosecond clocks: How long does a particle actually take to tunnel?

Part 1: The Wall That Is No Wall — Why Tunneling Is Possible

To understand why tunneling is permitted, we have to abandon one classical image: the idea that a particle is a tiny, clearly localized ball with a definite location. In quantum mechanics a particle is described by a wavefunction, whose squared magnitude tells us the probability of finding the particle at a given place. This wavefunction is not a substance that stops somewhere; it is a possibility smeared out over space.

Here comes the decisive point. When this wave meets a barrier — a region where the potential energy is higher than the particle's energy — the wavefunction does not vanish abruptly at the barrier's edge. It penetrates. Inside the barrier it does not oscillate but decays exponentially. If the barrier is thin enough, the wavefunction on its far side is greatly weakened but has not fallen to zero. And whatever is not zero means: there is a nonzero probability of finding the particle beyond the barrier. That, precisely, is tunneling. The particle was never "in" the wall in any classical sense; it simply reappears, with a certain probability, on the other side.

The mathematics behind it is astonishingly compact and, I am of the opinion, one of the most beautiful formulas in all of quantum mechanics. The probability \(T\) that a particle penetrates a barrier depends approximately on the parameters like this:

$$T \approx e^{-2\int \kappa\, dx}, \qquad \kappa = \frac{\sqrt{2m\,(V - E)}}{\hbar}$$

You don't have to solve this formula in detail to read its message. Three quantities decide the particle's fate, and all three sit under an exponent — meaning that small changes have dramatic consequences:

• The width of the barrier. The thicker the wall, the further the wave must decay, the less likely the crossing. Doubling the width does not halve the probability; it makes it exponentially smaller.
• The height of the barrier relative to the particle's energy (\(V - E\)). The higher the hill above the energy brought along, the steeper the decay.
• The mass of the particle. And this is perhaps the most surprising factor: tunneling is an effect of light particles. An electron tunnels a millionfold more easily than a proton, and a proton in turn more easily than a whole atom. This is why tunneling is ubiquitous in the world of electrons and atomic nuclei and utterly unmeasurable in the world of footballs. A macroscopic ball has such a large mass that the tunneling probability is a number with astronomically many zeros after the decimal point — you could wait for the age of the universe a billion times over and never see it pass through the wall.

This exponential sensitivity is the thread running through this entire article. It explains why tunneling either practically never happens or happens overwhelmingly often, why an instrument can resolve the spacing of individual atoms, and why swapping an isotope in biochemistry can suddenly change everything.

Part 2: The First Triumph — Gamow and Alpha Decay (1928)

The first great application of the tunneling idea was at the same time one of the earliest convincing proofs that the young quantum mechanics holds even at the heart of the atomic nucleus. Around 1928 nuclear physics faced a stubborn puzzle: alpha decay. Heavy radioactive nuclei such as uranium or radium occasionally hurl out an alpha particle — a bundle of two protons and two neutrons, that is, a helium nucleus. The problem was twofold and seemed insoluble.

First: if you measure the energy with which an alpha particle leaves the nucleus, and calculate how high the nucleus's electrical repulsion barrier is, you find that the particle should not be able to clear the barrier at all. Its energy is far too low. Classically, the alpha particle would have to remain trapped in the nucleus forever, like a ball in a crater whose walls are higher than the ball can ever jump.

Second: the physicist Hans Geiger and his colleague John Nuttall had already discovered, in 1911/1912, a curious empirical rule, the Geiger-Nuttall law. It links the half-life of an alpha emitter to the energy of the emitted particle — and across a breathtaking range: between the longest-lived and the shortest-lived emitters lie more than twenty orders of magnitude in decay rate, while the energy differs by only a few percent. A tiny change in energy leads to a gigantic change in lifetime. No one had an explanation for this extreme sensitivity.

Enter George Gamow, a young, unconventional Russian physicist who found the solution in the summer of 1928 — and independently of him, almost simultaneously, so did the pair Ronald Gurney and Edward Condon. Gamow's idea was as simple as it was bold: the alpha particle does not clear the barrier, it tunnels through it. It sits in the nucleus, slams against the inner wall of the repulsion barrier countless times per second, and on each attempt it has a tiny but nonzero chance of tunneling through. Multiply the tiny tunneling probability by the enormous number of attempts, and you obtain a decay rate — and hence a half-life.

The real triumph lay in the detail. Gamow treated the repulsion barrier approximately, piece by piece, as locally rectangular, defined a position-dependent decay rate at each point, and integrated across the entire "classically forbidden" zone — exactly the procedure known today as the WKB approximation. The result, the famous Gamow factor, depends exponentially on the reciprocal square root of the energy. And it is precisely this dependence that reproduces the Geiger-Nuttall law with its extreme sensitivity across more than twenty orders of magnitude. The tiny change in energy sits under an exponent — hence the huge effect on lifetime.

The significance of this moment can hardly be overstated. It was one of the first cases in which the new quantum mechanics did not merely explain the behavior of electrons in the atomic shell, but reached quantitatively and precisely — across twenty orders of magnitude — into the atomic nucleus itself. Tunneling was no longer a mathematical possibility; it was the mechanism behind the radioactivity one could measure in any laboratory. And the same radioactive clock that Gamow explained still provides, to this day, the basis for dating rocks, fossils, and the planet Earth.

Part 3: Why the Sun Shines — the Gamow Peak

If tunneling lets particles out of the nucleus, then conversely it can also bring together particles that ought to repel each other. And that is precisely what the existence of the Sun — and therefore our own — rests upon.

The Sun draws its energy from nuclear fusion: in its core, protons merge over several steps into helium, releasing enormous energy. But protons are both positively charged and repel each other ever more fiercely as they draw closer — the notorious Coulomb barrier. For two protons to fuse, they must approach each other to within the reach of the strong nuclear force, that is, overcome the Coulomb barrier. And here one runs into a problem that embarrassed the early astrophysicists: the temperature in the solar core is around 15 million kelvin. That sounds tremendous, but the thermal energy of the protons at this temperature is nowhere near enough to overcome the Coulomb barrier classically. If you work it out, the Sun ought not to fuse to any appreciable degree at its core temperature. Yet it does — otherwise we would not exist.

The rescue is, once again, tunneling. The protons do not leap over the Coulomb barrier; they tunnel through it. But this gives rise to a subtle and beautiful fine-tuning that bears the name Gamow peak. It emerges from the struggle of two opposing tendencies:

• The Maxwell-Boltzmann distribution of particle energies falls off steeply toward high energies. There are very many protons with low energy and very few with high energy. If you wanted to rely on the fastest, most energetic protons, there would be almost none of them.
• The tunneling probability, by contrast, rises sharply with energy. The more energy a proton brings, the more easily it penetrates the barrier. The sluggish, slow protons practically never tunnel.

Multiply these two curves — the many slow particles with their vanishing tunneling chance and the rare fast particles with their good tunneling chance — and there remains a narrow energy window in which both factors are simultaneously large enough. This window is the Gamow peak. It lies far above the mean thermal energy but far below the height of the Coulomb barrier. In precisely this window fusion happens. The Sun burns neither with its typical particles nor with its most energetic ones, but with a rare, privileged minority that is fast enough to have a decent tunneling chance and common enough to occur at all.

This has a consequence that reaches far beyond physics. Because the fusion rate depends so sensitively on this tunneling window, the Sun burns extraordinarily slowly and steadily — over billions of years instead of in a cosmic explosion. The sheer sluggishness of the tunneling process is the reason stars live so long and why there was time enough for life to arise. The same fusion that builds the heavy elements inside stars, and whose most spectacular chapter we examined in the article on Cosmic Gold: Neutron Star Collisions, the r-Process, and the Origin of the Heavy Elements, begins with this quiet, improbable leap through a wall that ought to be impenetrable.

Part 4: From Riddle to Tool — the Tunnel Diode and the Microscope That Sees Atoms

For half a century tunneling was mainly an explanation for natural phenomena: for radioactivity, for the glow of the stars. But from the 1950s onward it transformed from an explained effect into a mastered tool. Two inventions, both crowned with the Nobel Prize, exemplify this turn.

The first came from Leo Esaki, a Japanese physicist who in 1957/1958, while studying heavily doped semiconductors, stumbled upon a strange behavior. In a special, very thin semiconductor diode — today called the Esaki or tunnel diode — electrons could tunnel directly through the junction rather than over it. This led to a paradoxical effect: within a certain voltage range the current dropped when the voltage was increased — a so-called negative differential resistance, which classically should not exist. This property made the tunnel diode an extraordinarily fast component, long used in high-frequency circuits. Above all, though, it was the first tangible proof that tunneling could not merely be observed but deliberately built into a component and put to technical use. For this Esaki received the 1973 Nobel Prize in Physics — together with Ivar Giaever and Brian Josephson, whose work concerned the tunneling of electrons and electron pairs in superconductors. Josephson's tunneling effect, the coherent tunneling of Cooper pairs through a thin barrier, remains to this day the basis of the most sensitive magnetic-field sensors and of superconducting qubits — a thread we took up in detail in the article on Resistance Is Futile: Superconductivity from Onnes to BCS to the Room-Temperature Dream.

The second invention is perhaps the most spectacular application of tunneling of all. In 1981 Gerd Binnig and Heinrich Rohrer built, at the IBM laboratories near Zurich, the scanning tunneling microscope (STM). The idea exploits the exponential sensitivity of tunneling in an ingenious way. You bring an extremely fine metal tip — ideally ending in a single atom — to within a few atomic diameters of a conducting surface, so close that electrons tunnel across the tiny vacuum gap between tip and surface and produce a measurable tunneling current. And because this current depends exponentially on the distance, it changes by about an order of magnitude as soon as the gap narrows by the diameter of a single atom.

This brutal sensitivity is the whole secret. If you raster the tip line by line across the surface and keep the tunneling current constant by adjusting the height, then the up-and-down motion of the tip traces a map of the surface — with a resolution that makes individual atoms visible. For the first time in history, humans could not merely calculate that atoms exist but image them directly and, as soon became clear, even move them one at a time. Binnig and Rohrer received the Nobel Prize in Physics for this as early as 1986 — remarkably quickly. The STM and its descendants became an indispensable foundation of nanotechnology, from semiconductor research through metallurgy to molecular biology.

Part 5: The Tunneling in Your Pocket — Flash Memory

There is a good chance that right now you are carrying a device whose function rests directly on quantum tunneling: a USB stick, an SSD, the memory module of your smartphone. Modern flash memory stores each bit as an electric charge on a tiny, all-around-insulated conductor, the so-called floating gate. If the floating gate is charged, the bit reads one value; if uncharged, the other. The insulation all around ensures that the charge is preserved there for years even without a power supply — which is why the USB stick keeps its data when you pull it out of the computer.

But how does one get the charge onto an all-around-insulated gate in the first place, and how does one get it off again? Classically this would be impossible — the insulating layer exists precisely to let no charge through. The answer, again, is tunneling. When programming and erasing, you apply a strong voltage that makes the electrons tunnel through the merely one-to-two-nanometer-thin insulating layer — a process known as Fowler-Nordheim tunneling. Because the electrons pass through the barrier rather than over it, an astonishingly moderate energy suffices.

Here the exponential sensitivity of tunneling shows its most useful and, at the same time, most delicate side. It allows deliberate writing and erasing at a reasonable voltage. But the same sensitivity sets a hard limit on miniaturization: if the insulating layer becomes too thin or neighboring memory cells crowd too closely together, electrons tunnel even when they should not — the charge leaks away, data decay, cells interfere with one another. It is precisely to this limit that we owe a large part of the engineering craft behind modern memory, such as the transition from flat to three-dimensionally stacked memory cells. The effect that makes flash memory possible in the first place is at the same time the enemy that halts its shrinkage. I am of the opinion that there is hardly a finer example of how a single quantum effect can simultaneously sustain and constrain an entire industry.

Part 6: The Tunneling in the Living — Enzymes and the Isotope Fingerprint

For a long time tunneling was regarded as something that happens in stars, atomic nuclei, and cold semiconductor laboratories — far from the warm, wet, chaotic world of biology. This view has shifted over the past two decades. Today it is widely accepted that enzymes — the molecular catalysts of life — exploit tunneling in certain reactions, particularly in the transfer of hydrogen.

The reason lies, once more, in mass. Hydrogen is the lightest of all atoms, and its nucleus, the proton, is so light that its wavefunction is appreciably "smeared out." When an enzyme transfers a hydrogen atom from one molecule to another, this light particle must overcome an energy barrier. And instead of laboriously climbing over it, it tunnels through to a considerable extent. For the much heavier carbon, nitrogen, or oxygen atoms this would be unthinkable; for hydrogen it is an everyday occurrence.

How do we know this? The most elegant proof is the kinetic isotope effect. In a substrate you replace ordinary hydrogen with its twice-as-heavy isotope deuterium (a proton plus a neutron). For classical chemistry this should change the reaction rate only moderately. But in tunneling, mass hits exponentially: the twice-as-heavy deuteron tunnels far worse than the proton, and the reaction becomes dramatically slower — often more so than any purely classical barrier could ever explain. An unusually large isotope effect is therefore a fingerprint of tunneling. Newer research, however, paints a subtler picture: in 2024 a reaction model was described that proceeds exclusively via hydrogen tunneling and yet shows only a small isotope effect — an indication that the environment of the tunneling particle (for instance, coupling to the solvent) can soften the simple relationship between a large isotope effect and tunneling. The details of biological tunneling are thus still the subject of active research.

Nevertheless, the broad picture remains remarkable: an effect that Gamow conceived to explain the decay of uranium is found today at the very core of our metabolism. The enzymes that keep us alive — a field we examined elsewhere using the example of The Machine That Splits Water: Photosystem II and the Origin of Oxygen — use the same quantum rule as the core of a star.

Part 7: The Hardest Question — How Long Does a Leap Through the Wall Take?

We have now seen that tunneling happens and how often. There remains a question that sounds harmless and yet has divided physics for nearly a century: How long does tunneling take? When a particle goes in on one side of the barrier and reappears on the other — how much time does it spend "in" the wall? Does it take any time at all? Or does the crossing happen instantaneously?

The question is trickier than it looks. Classically you could simply start the clock at entry and stop it at exit. But in quantum mechanics the particle inside the barrier has no well-defined location and no well-defined velocity; the classically forbidden zone is exactly the region about which our intuition falls silent. Over decades theorists proposed various, mutually contradictory definitions of a "tunneling time," and some of them even led to the astonishing statement that the particle could cross the barrier faster than the speed of light — which, on closer analysis, does not, however, mean that information could be transmitted faster than light.

Only in recent years did the question become experimentally tractable, and it did so with one of the most ingenious measuring instruments in modern physics: the attoclock. An attosecond is the billionth part of a billionth of a second — an interval so unimaginably short that light travels only the diameter of a small molecule in one attosecond. The attoclock's principle is elegant: an extremely short, circularly or elliptically polarized laser pulse tears an electron out of an atom by letting it tunnel through the atom's Coulomb barrier. Because the laser's electric field rotates like a racing hand as it does so, the tunneled-out electron is deflected in a different direction depending on the instant of its exit. The angle at which the electron ultimately flies away thus acts like the position of a hand on a dial — it encodes the time that elapsed between the beginning and the end of the tunneling event.

The result of these experiments, carried out in ever more refined form since about 2014, is as surprising as it is contested. Several works concluded that the tunneling time is vanishingly small, possibly exactly zero: the electron seems not to linger any measurable moment in the barrier but to appear on the other side without appreciable delay. Other groups disagreed and measured nonzero times. The dispute arises because the analysis requires delicate theoretical assumptions: the interplay of the laser field and the atomic core's own electrical attraction cannot be cleanly disentangled without contested approximations, and depending on which definition of "tunneling time" you adopt, you obtain different answers. A widely noted review from 2020 weighed the evidence and saw the balance tip toward the "zero-time" scenario; precision measurements up to 2022 refined the picture further without settling the debate for good.

I am of the opinion that it is precisely this open question that makes tunneling so fascinating. Nearly a hundred years after Gamow, with applications from stellar physics to the USB stick, we still do not fully understand what a particle does while it is in the forbidden zone — or whether the question of "duration" even has a physically well-defined meaning. It is a salutary reminder that even a textbook effect still harbors real, unsolved physics at its edges.

A Framework: What Governs Tunneling

If you want to bring the many faces of tunneling to a common denominator, it is the exponential sensitivity to a few parameters. The following table summarizes which quantity is decisive in which example:

The common principle: because all these quantities sit under an exponent, a small difference in the setup makes an enormous difference in the outcome. Therein lies both the power and the treachery of the effect — it enables instruments of atomic sharpness and at the same time sets a hard limit on the shrinkage of our chips.

The Central Takeaway

Perhaps the most important lesson of quantum tunneling is one about the nature of limits. In our classical intuition a barrier is absolute: whatever lacks enough energy does not get through, and that's the end of it. Quantum mechanics replaces this all-or-nothing with a probability. The wall is no longer impenetrable, only more or less likely to be penetrated — and this probability responds hypersensitively to the circumstances. The forbidden does not become permitted, but it becomes rarely possible, and from this rarity everything else follows: the patient, million-year fusion of the stars, the statistical clock of radioactive decay, the atomic sharpness of a microscope.

For the practically minded person — whether engineer, scientist, or simply a curious mind — there is a transferable thought in this: some barriers you overcome not with brute force but by changing the conditions so that the improbable crossing becomes more probable. You make the wall thinner, you lower its height, you increase the number of attempts. The exponential lever works in both directions. Whoever knows it understands why an extra nanometer of insulating layer decides between working and defective memory — and why a single swapped neutron in a hydrogen atom slows a biochemical reaction manyfold.

And finally, the still-open question of tunneling time teaches us humility. We command tunneling technically down to the chips in our telephones, and yet the seemingly simple question of how long the leap through the wall takes eludes us. The quantum world can be used long before it can be fully understood.

Reflection Question

Tunneling shows that a limit which appears absolute may in truth be only a matter of probability. Where in your own thinking — about technology, about risks, about what is "impossible" — do you still treat a barrier as an absolute wall, when it may be only a very improbable but not excluded passage? And what would change if you stopped asking "is it possible or not?" and instead asked "how do I make the crossing more probable?"

Cross-References in the Vault

• Resistance Is Futile: Superconductivity from Onnes to BCS to the Room-Temperature Dream – the Josephson effect as coherent tunneling of Cooper pairs, honored in the same 1973 Nobel Prize as Esaki's tunnel diode.
• Order from Noise: Quantum Error Correction and the Road to a Fault-Tolerant Quantum Computer – how to master macroscopic quantum states, kin to the art of putting tunneling to deliberate use.
• Spooky Action at a Distance: Quantum Entanglement from Einstein to the Quantum Internet – another case in which quantum mechanics openly contradicts classical intuition.
• Cosmic Gold: Neutron Star Collisions, the r-Process, and the Origin of the Heavy Elements – the fusion in stars that is set in motion only by the tunneling effect.
• The Machine That Splits Water: Photosystem II and the Origin of Oxygen – enzymes as molecular machines in which hydrogen tunneling plays a role.

Sources

• Merzbacher, E. et al.: The Early History of Quantum Tunneling. Physics Today. https://physicstoday.aip.org/features/the-early-history-of-quantum-tunneling
• 4.5: Alpha Decay (Gamow theory, Geiger-Nuttall law). Physics LibreTexts. https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/04:_One-Dimensional_Potentials/4.05:_Alpha_Decay
• Gamow Explains Radioactive Alpha Decay with Quantum Tunneling. EBSCO Research Starters. https://www.ebsco.com/research-starters/history/gamow-explains-radioactive-alpha-decay-quantum-tunneling
• Gamow Peak / stellar nuclear fusion. Bohrium Sciencepedia. https://www.bohrium.com/en/sciencepedia/feynman/keyword/gamow_peak
• Press release: The 1986 Nobel Prize in Physics (scanning tunneling microscope, Binnig & Rohrer). NobelPrize.org. https://www.nobelprize.org/prizes/physics/1986/press-release/
• The Invention of Tunneling-Based Flash Memory (Fowler-Nordheim tunneling). 3D InCites. https://www.3dincites.com/2018/02/invention-tunneling-based-flash-memory/
• Sabbah, A. J. et al.: The attoclock and the tunneling time debate. J. Phys. B (IOPscience). https://iopscience.iop.org/article/10.1088/1361-6455/ab6b3b
• Full experimental determination of tunneling time with attosecond-scale streaking method. Light: Science & Applications (2022). https://www.nature.com/articles/s41377-022-00911-8
• Roque, J. et al.: Hydrogen Tunneling Exhibiting Unexpectedly Small Primary Kinetic Isotope Effects. Chemistry – A European Journal (2024). https://chemistry-europe.onlinelibrary.wiley.com/doi/10.1002/chem.202401323

Note: The tunneling-time question is an actively contested field of research; the tendency toward a "vanishing" tunneling time mentioned in the text reflects the state of a widely noted review and should not be understood as settled consensus.

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