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§ Phase vi · Tunneling · Chapter 01

Through the barrier

Roll a marble at a hill it cannot climb and it rolls back, every time. Schrödinger's equation, asked the same question about an electron in 1927, refused to give the same answer. A small but non-zero piece of the wavefunction kept slipping through, as if a part of the particle had simply ignored the wall. The history of solid-state electronics, of stellar fusion, and of the bomb is downstream of that small piece.

There is a feeling, when you first see the math of quantum tunneling, that you have caught the universe cheating. The setup is so familiar: a particle of energy E moving toward a wall of height V₀, with E less than V₀. Classical mechanics says one thing and says it with absolute confidence. The particle reaches the wall, runs out of kinetic energy, and turns around. It comes back. It always comes back. You can repeat the experiment a million times with billiard balls or rolling marbles or planets caught in a gravitational well, and the answer is the same. Insufficient energy means no crossing. This is one of the first things you learn in any introductory mechanics course, and for two and a half centuries no one had any reason to doubt it.

Then in early 1927, with Erwin Schrödinger’s wave equation barely a year old, a young physicist named Friedrich Hund sat down to study what happens when you write the same problem in wave language. The atom of intuition he picked was a double-well potential, two valleys separated by a hump, as a model for an ammonia molecule whose nitrogen can sit on either side of the hydrogen triangle. He expected the math to confirm what physical chemistry already knew: that a molecule stuck in one well stays there. The math told him otherwise. The wavefunction did not stop at the hump. It leaked through. And the leak was small but finite and predictable, controlled by the height and width of the barrier in a way Hund could write down in closed form.

Within a year the same equation had been pointed, by George Gamow in Göttingen and independently by Ronald Gurney and Edward Condon in Princeton, at the most spectacular barrier in physics: the wall of an atomic nucleus. Out of it came alpha particles. The classical picture had no idea how they got out, because they had too little energy to climb over. The quantum picture said they did not climb. They tunneled. Gamow’s paper, written in a few weeks during a summer visit, predicted the lifetime of every alpha emitter in the periodic table from one tidy formula. The match to experiment ran across forty orders of magnitude, from microseconds to billions of years. Tunneling went from an oddity in a wave equation to a hard quantitative tool almost overnight.

This chapter is about how that works, beat by beat. Not the nuclear case yet (that is the next chapter), but the simplest version: a particle, a rectangular barrier, and the question of what the wavefunction does inside a region where classical physics forbids it to be. Once you see that piece, alpha decay, scanning microscopes, flash memory, and the cold light of stars all become applications of one short calculation.

The cleanest setup, the one every textbook uses because the algebra is short enough to fit on one page, is a rectangular barrier. Outside the barrier the potential is zero. Inside, for a width L, the potential is a flat plateau of height V₀. The particle has total energy E, and we restrict ourselves to the case E less than V₀, because that is where classical and quantum disagree. In the two outside regions the particle is free, and Schrödinger’s equation reduces to the same wave equation that describes light in vacuum or sound in air. The solutions are sinusoids. Write them as e^{ikx} for waves moving right and e^{-ikx} for waves moving left, with the wavenumber k tied to the energy by E = ℏ²k²/(2m). This is just the de Broglie relation in its plainest form, and it tells you the particle on the left has a well-defined wavelength set by its speed.

Now ask what the equation says inside the barrier. Schrödinger’s equation in its time-independent form reads −(ℏ²/2m) · d²ψ/dx² + V(x) · ψ = E · ψ. Inside the barrier V(x) = V₀, and you can rearrange the equation to d²ψ/dx² = (2m(V₀ − E)/ℏ²) · ψ. The coefficient on the right is positive because V₀ exceeds E. Compare this to the free equation, where the coefficient is negative and the solutions are sines and cosines. With a positive coefficient the solutions are no longer oscillating; they are real exponentials. Define κ = √(2m(V₀ − E))/ℏ, a real positive number with units of inverse length, and the wavefunction inside the barrier becomes a sum of e^{+κx} and e^{−κx}. The piece you might worry about is the growing exponential e^{+κx}, but for a finite barrier of moderate width both pieces are present and matched at the boundaries by continuity. The dominant behavior is decay. The wavefunction does not vanish at the wall. It bends and falls, slowly if V₀ − E is small and quickly if it is large. By the time it reaches the far side of the barrier it has been reduced by roughly e^{−κL}, where L is the width.

The fact that the wavefunction has nonzero amplitude on the far side of the barrier is what we call tunneling. The word is borrowed and slightly misleading. There is no tunnel. There is no hole. The particle does not bore a path through the wall the way a worm bores through an apple. The wavefunction simply has support in the forbidden region, and the support extends all the way through. If you put a detector on the right side of the barrier, you will sometimes see the particle there, and when you do, you will measure it with its original energy E intact. Energy is conserved exactly. Nothing has been borrowed and nothing has been paid back. The only thing that has changed is your conditional probability of where to look for the particle.

The size of the transmitted piece is the practical question. Matching the incoming, decaying, and transmitted pieces at the two boundaries (requiring ψ and its derivative to be continuous) gives a closed-form expression for the transmission coefficient T, the fraction of the incoming probability current that makes it through. For a thick barrier, where κL is comfortably larger than 1, the result simplifies to a clean formula. The transmission probability falls off as e^{−2κL}, with κ controlled by how far the particle’s energy E sits below the barrier height V₀. Double the barrier width and you square the suppression. Halve the energy deficit V₀ − E and you double the exponent. The dependence is exponential, which is why tunneling probabilities can vary across forty orders of magnitude between one alpha-emitting isotope and the next, even when their physical sizes differ by less than a factor of two. Tiny changes in the exponent become enormous changes in the rate.

Notice that the formula has no notion of a sudden break at E = V₀. Classically there is a cliff. Below the cliff, transmission is zero; above the cliff, transmission is one. The quantum curve has no cliff. Below V₀ the transmission is small but nonzero and growing; at V₀ the transmission is already substantial; above V₀ the transmission climbs smoothly toward unity, with small oscillations due to wave interference inside the barrier region (the same Fabry-Pérot oscillations you would see in a thin film of glass on a window). There is no point at which a quantum particle “suddenly starts” passing through. The transmission is a smooth function of energy from the bottom of the well to the top of the barrier and onward. The all-or-nothing classical answer is replaced by a gentle, continuous probability curve.

This is worth holding onto, because it explains a great deal about why electronics works the way it does. A flash memory cell stores a bit of information by trapping electrons on a tiny floating gate, isolated from the rest of the transistor by a thin oxide barrier. To write the bit you raise the voltage and persuade the electrons to tunnel onto the gate; to erase it you flip the voltage and persuade them to tunnel off. The energy barrier is fixed by the material; only the width of the effective barrier changes with voltage. Because the transmission depends exponentially on barrier width, a small voltage change produces an enormous change in the writing rate, which is exactly the nonlinearity you want for digital memory. The whole industry of nonvolatile storage rides on this single exponential.

Region I  (x < 0):       ψ_I  = A · e^{+ikx} + B · e^{-ikx},      k = √(2mE)/ℏ
Region II (0 < x < L):   ψ_II = C · e^{+κx} + D · e^{-κx},        κ = √(2m(V₀−E))/ℏ
Region III (x > L):      ψ_III = F · e^{+ikx}

F / A = 4 i k κ · e^{-ikL} / [ (κ + ik)² · e^{-κL} − (κ − ik)² · e^{+κL} ]

T(E) = 1 / [ 1 + (V₀² · sinh²(κL)) / (4 · E · (V₀ − E)) ]

T(E) ≈ 16 · E · (V₀ − E) / V₀² · e^{-2κL}

The double-barrier resonance is a genuinely beautiful piece of physics, and worth one more figure to see clearly. If you put two thin barriers in series with a small well between them, the well has its own quasi-bound states with discrete energies, just like a finite square well. When the energy of the incoming wave matches one of those quasi-bound energies, something remarkable happens. The wave entering the well builds up by constructive interference. The amplitude inside the well grows until the probability of leaking out the far side becomes equal to the probability of having entered, and the transmission coefficient climbs to one. It is the quantum-mechanical version of the same trick that lets a Fabry-Pérot etalon transmit a sharp line of light through two mirrors that are individually almost opaque. The double-barrier transmission curve has sharp peaks at the resonance energies, and between them the suppression is worse than a single barrier (you have to tunnel through two of them). It is a comb of bright lines on a dark background, and you can engineer where the lines sit by changing the well width.

The story of how this single piece of math worked its way through twentieth-century physics is one of the great hot streaks in modern science. Hund used it in 1927 for ammonia inversion. Gamow used it in 1928 for alpha decay and explained, in a single paper, why uranium-238 has a half-life of 4.5 billion years and polonium-212 has a half-life of 0.3 microseconds, two numbers separated by a factor of 5 × 10^{23} that fall out of the same formula evaluated at slightly different energies. The same formula explains cold electron emission from sharp metal tips, which is the basis of the field-emission electron microscope. It explains why the Sun shines: two protons in the solar core do not have enough thermal energy to overcome their mutual electrostatic repulsion, but the wavefunction of one extends just barely into the forbidden region around the other, and the tiny e^{−2κL} probability of tunneling, multiplied by the absurd number of collisions per second in a stellar core, is just enough to set the fusion rate that has kept the Sun warm for 4.6 billion years. Without quantum tunneling, the Sun would be cold.

Leo Esaki put the formula to industrial use in 1957 by deliberately engineering a semiconductor diode whose I-V curve had a region of negative differential resistance, courtesy of tunneling between two heavily doped layers. He shared the 1973 Nobel Prize with Ivar Giaever and Brian Josephson for that and related work. Binnig and Rohrer turned the exponential sensitivity of tunneling current to tip-sample distance into an imaging technique sharp enough to see single atoms; their 1981 scanning tunneling microscope won the Nobel Prize five years later. The 2025 Nobel Prize, awarded as this book is being written, recognized Clarke, Martinis, and Devoret for showing that the same effect operates on macroscopic Cooper pairs in superconducting circuits, scaling tunneling from one electron to a flowing condensate. The same e^{−2κL}, the same κ, the same exponential decay inside a forbidden region. The clarity of the original 1927 picture has not faded.

The point to carry forward, before we leave the rectangular barrier behind and turn to the real curved barrier of a nucleus, is that tunneling is not exotic. It is not a strange exception to ordinary quantum mechanics. It is the same wave equation, applied to a region where the potential exceeds the energy, giving the same kind of answer it always gives: a smooth solution that respects all the boundary conditions. The wave equation does not know the difference between “classically allowed” and “classically forbidden.” It has no concept of an energy threshold below which a particle is required to stop. It only knows about curvature and continuity. In one region the curvature of ψ points one way and you get sines and cosines. In another the curvature points the other way and you get exponentials. The two solutions match at the boundary because they must, and the matching forces a small but nonzero amplitude through whatever wall you have set up. The rest is bookkeeping.

What this opens up is a class of phenomena that classical physics simply cannot reach. Stars cannot fuse without it. Radioactive nuclei cannot decay by it. Field-effect transistors cannot leak without it. Microscopes cannot image atoms without it. In each case the central piece of the calculation is the same exponential factor, and the work is in deciding what the barrier looks like, how broad it is, and what particle is doing the tunneling. The next chapter takes the picture we have built here and applies it, with the small adjustment of a curved Coulomb barrier instead of a rectangular one, to the first dramatic application: the alpha particle escaping the uranium nucleus, and Gamow’s beautiful prediction of the half-lives of every heavy element in the periodic table.