The Wall Is a Change of Sign

Lesson · Mathematics · Part Ten

The Wall Is a Change of Sign

Classical mechanics forbids the crossing for an exact reason: inside the barrier there is no real number left to be the momentum. The Schrödinger equation does not stop there. One coefficient changes sign, the solution stops turning in space and starts dying instead, and a faint tail reaches the far side. The Sun runs on this. So does the microscope that first imaged atoms. Nothing burrows, nothing borrows, nothing outruns light.

John Rector 26 August 2026 ~15 min For philosophers and theologians

Contents

The wall, according to classical mechanics
One equation, one sign
Inside the wall
The signature is exponential
Three things tunnelling does not license
The direction of explanation
The ledger
Exercises

01 The wall, according to classical mechanics

A bowl, a marble

Roll a marble up the inside of a bowl. It climbs, slows, stops, and comes back. The height where it stops is not decoration; it is bookkeeping. The marble arrived with a fixed budget of energy, every centimetre of height has a price, and the marble climbs exactly until the budget is spent — not a millimetre past.

Classical mechanics does not merely predict the turnaround. It has nothing else it could say.

Here is the prohibition, stated so that it can be inspected. A particle with a fixed total energy E moves through a landscape whose height at each point costs potential energy V. Whatever is left over is kinetic — energy of motion — and kinetic energy is ½mv²: a mass times the square of a speed. That square is the entire argument.

p² = 2m(E − V) ≥ 0 the classical books — momentum squared is never negative, because squares of real numbers are never negative

So wherever the landscape rises above the budget — wherever V > E — the momentum would have to be the square root of a negative number. And classical mechanics has one commitment it never negotiates: every physical quantity is a real number. For a particle of that energy in that landscape, the high ground is not hard to reach. It is not in the theory. There is no classical state with that position and that energy; the equations of motion simply have no solution there. “Forbidden” is not a fence. It is an absence.

Notice what kind of moment this is, because the series has been here before. A square root goes negative, and a theory has two options. Part One watched mathematics face this exact fork at √−1: declare the territory closed, or admit the number system was too small. Classical mechanics chose the first. We are about to watch what happens when a theory chooses the second.

02 One equation, one sign

Quantum mechanics replaces the marble’s position with a state. For a particle of definite energy E, the spatial profile of that state is a function ψ(x), and it obeys the time-independent Schrödinger equation. Every symbol in it is already on the table: m the mass, V(x) the landscape, ħ Planck’s constant divided by 2π, and ψ″ the second derivative — the curvature of the graph of ψ.

ψ″ = (2m/ħ²) (V − E) ψ read it as an instruction for bending: the coefficient tells the graph which way to curve, and how hard

Everything in this lesson is the sign of that one coefficient.

E > V: ψ″ = −k²ψ, k = √(2m(E−V))/ħ → e^±ikx — the solution turns E < V: ψ″ = +κ²ψ, κ = √(2m(V−E))/ħ → e^±κx — the solution dies or blows up both k and κ are real numbers — the imaginary unit lives in the exponent on one side and not the other

You have seen this dichotomy before, wearing other clothes. It is the trichotomy of Part Six and the “I Want” essay: a rate x′ = kx with an imaginary rate turns in a circle; with a real rate it grows or decays along a ray. Where the particle is allowed, the exponent’s rate is ik — imaginary, so the solution rotates through the complex plane as you move along x, which is what “oscillation” is. Where the particle is forbidden, replace k by iκ and the two i s annihilate: e^ikx → e^−κx. The quarter-turn generator becomes a plain decay rate. The wall is the place where the equation stops turning.

A careful reader — and this series assumes no other kind — should object here. The Schrödinger equation is second-order; the trichotomy was stated for a first-order rate. The repair is short and it is rigorous. Track the pair (ψ, ψ′) instead of ψ alone. The second-order equation is exactly a first-order equation for that pair, and the matrix driving it has eigenvalues ±ik in the allowed region — purely imaginary, so the pair genuinely rotates, tracing an ellipse — and ±κ in the forbidden region — real, so the pair grows and decays along hyperbolas. Equivalently: ψ″ + k²ψ factors as (d/dx − ik)(d/dx + ik)ψ, two first-order turnings whose solutions together span everything the second-order equation can do. The aphorism survives inspection; it just needed the phase plane to stand on.

Two boundary flags, posted now rather than in the small print. First, exactly at the wall’s energy — E = V — the coefficient is zero, and ψ″ = 0 has the affine solutions a + bx: straight-line drift, not stillness. The marginal case is its own third thing. Second, and more important: the turning that stops is the turning in space. The full state is ψ(x) e^−iEt/ħ, and that time factor — the engine of Part Three’s whole ladder — keeps rotating everywhere, barrier included. Inside the wall the state still turns in time; what it has lost is its rotation along x. Keep the aphorism spatial and it is exact.

Figure 01

One equation, two behaviours — the sign of V − E decides

The same equation, plotted on the two sides of the sign. Both curves are computed, not sketched: the left panel is the cosine drawn as Bézier half-periods, the right is e^−x/λ sampled every 4 units. In the phase plane the left pair (ψ, ψ′) traces an ellipse — genuine rotation — while the right runs along a hyperbola.

The wall is not where the world ends. It is where one coefficient crosses zero and the solution changes key — from turning to dying.

03 Inside the wall

Now build the actual object. Send an electron with E = 5 eV at a barrier of height V₀ = 6 eV and width 3 Å — the scale of a few atoms, the working conditions of a scanning tunnelling microscope. On each side of the barrier the solution turns; inside it, it dies. Quantum mechanics demands only that the pieces join smoothly — the function and its slope continuous at both faces — and that stitching requirement determines everything: how much reflects, how much survives, and the exact shape of the corpse in between.

Figure 02

The stitched solution: turn, die, turn again — smaller

Re ψ where the particle is allowed
Re ψ inside the barrier
envelope |ψ| (left of the wall)

The matched solution, computed exactly: electron at E = 5 eV, barrier V₀ = 6 eV, width 3 Å. Wavelength 5.48 Å on both sides — energy is conserved, so the transmitted wave turns at exactly the incident rate, only smaller. Decay length inside: 1/κ = 1.95 Å. Transmission T = 0.1015, reflection R = 0.8985; they sum to 1 to the last digit, which is the check that catches sign errors — it caught two of mine while this figure was being computed.

Look at what lives inside the wall. Not nothing — the function is not zero there. Not oscillation either: no nodes, no wavelength, just a sagging exponential. The state persists where the classical particle could not exist, but persists in a different key. And its exponent invites a dangerous thought: formally, e^−κx responds to the momentum operator of Part Four with p̂ψ = iħκ ψ. Imaginary momentum?

No — and the rule from Part Four holds without exception: every measured momentum is real. The momentum operator’s spectrum — the complete list of values a measurement can return — is the real line. iħκ is not on the list, and the decaying exponential clipped to a barrier is not a legitimate eigenstate of momentum at all. The imaginary value is a property of the exponent, and exponents are not measurement outcomes. Inside the wall, the ghost axis is doing all the work precisely because no instrument is reading it.

Which sets up the trap a good student should now spring: catch the particle inside the wall. Detect it mid-barrier, weigh its kinetic energy, and read off the impossible number — E − V₀, which is negative. The books must show the deficit somewhere. Mustn’t they?

The answer has a theorem on top and a mechanism underneath, and it is worth keeping the two layers labelled. The theorem: kinetic energy is the operator p̂²/2m, a square, and a square is a non-negative operator — no measurement of kinetic energy ever returns a negative value. Full stop; nothing further is needed. (And “the kinetic energy at position x” — both quantities sharp at once — is not an observable of the theory at all; that is the commutator of Part Seven talking.) The mechanism, which is an estimate rather than a derivation but shows how the theorem is enforced: to find the particle inside the wall you must localise it within the penetration depth, Δx ≲ 1/κ. Kennard’s inequality then forces a momentum spread Δp ≳ ħκ — which is an energy uncertainty of order ħ²κ²/2m = V₀ − E. The deficit you hoped to observe is exactly the size of the disturbance the observation requires. The books do not balance by luck; they balance by construction.

One more thing, and Part Nine already paid for it: detecting the particle inside the barrier is a measurement update. It prepares a new state and begins a different experiment. It is not a peek at what the transmitted particle was doing on its way through — there is no formalism-level fact of that kind to peek at, a point section 05 will need again.

04 The signature is exponential

How much gets through? For a thick barrier the exact formula collapses to one term that matters:

T ≈ 16 (E/V₀)(1 − E/V₀) e^−2κa a is the barrier’s width — the prefactor is furniture; the exponential is the claim

An exponential in the width. That functional form is tunnelling’s fingerprint, and it is the reason the effect is simultaneously invisible in daily life and the working principle of half of modern instrumentation. Double a barrier that already eats a factor of a thousand and it eats a factor of a million. Shave an ångström off it and the current jumps an order of magnitude.

Figure 03

Transmission against width: a straight line on a log scale is an exponential’s confession

Computed from the exact rectangular-barrier formula for an electron 4.5 eV below the barrier top (a typical metal work function): the slope is −0.94 decades per ångström, a factor of about 9 per Å. This is the scanning tunnelling microscope’s entire trick — the tip never touches the surface; the exponential converts sub-ångström changes in gap into order-of-magnitude changes in current, which is why Binnig and Rohrer’s 1981 instrument could image individual atoms and earned the 1986 Nobel Prize, shared with Ruska.

Now point the same exponential at the sky. The Sun’s core sits near 15.7 million kelvin, which sounds enormous and is thermally pathetic: it corresponds to a typical thermal energy of 1.35 keV. For two protons to touch — to close within the one-or-two femtometres where nuclear attraction takes over — they must climb a Coulomb repulsion of roughly 1.4 MeV. That is about a thousand times the thermal budget. The fraction of proton pairs whose relative energy clears the barrier outright is of order e⁻¹⁰⁰⁰ ≈ 10⁻⁴³⁴, and the Sun owns only about 10⁵⁷ protons. Classically, the count of fusing pairs is zero. Not rare. Zero.

The Sun shines anyway because the barrier does not need to be cleared; it needs to be pierced. Run the competition between the Boltzmann tail (favouring the rare hot pairs) and the tunnelling exponent (forgiving the barrier faster as energy rises) and the product peaks near 6 keV — the Gamow peak, computed from the proton–proton Gamow energy of 493 keV — where the penetration factor is about 10⁻⁴ per encounter: absurdly generous compared to 10⁻⁴³⁴. One honesty clause: tunnelling explains how protons ever get within range; the Sun’s ten-billion-year patience is rationed by a separate bottleneck, the weak interaction that must convert a proton to a neutron in the same encounter. Tunnelling opens the door. It does not also cook the meal.

The oldest confirmation is the one that started the story. Radioactive nuclei spit alpha particles with energies that vary by barely a factor of two — yet their half-lives range from microseconds to ages older than the universe. Geiger and Nuttall tabulated that absurdity as an empirical rule in 1911–12, and it hung there, unexplained, until 1928. An alpha particle rattling inside a nucleus assaults the Coulomb wall on the order of 10²¹ times per second (a deliberately crude semiclassical figure), each attempt tunnelling with a probability set by that exponential — and an exponent that moves gently with energy moves the half-life by mountains.

Figure 04

Geiger–Nuttall: a factor of two in energy, twenty-four powers of ten in half-life

Seven alpha emitters, decay energies Q from 4.08 to 8.95 MeV (Q-values throughout — the emitted alpha carries slightly less, the recoiling nucleus takes the rest), half-lives from standard nuclear data. The dashed line is the least-squares fit log₁₀ t½ = 154.8/√Q − 59.0, computed for this figure. Gamow’s 1928 tunnelling integral predicts exactly this form. The vertical span is 10^24.2.

And light does it too — which is the detail that gives the game away. Press two glass prisms almost together. Beyond the critical angle, light should reflect totally off the first glass–air surface; Newton, in the Opticks, reported that if the second surface is brought close enough, light crosses the gap it had no right to enter and continues into the second prism — in his words, it “will go through that Surface, and through the Air or Vacuum between the Glasses, and enter into the second Glass.” Run the mathematics of light past the critical angle and you find the identical structure: the transverse wavevector goes imaginary, the wave in the gap is a decaying exponential, and transmission falls exponentially with gap width. Maxwell’s equations are classical; no ħ appears anywhere. What Schrödinger’s equation and Helmholtz’s share is not physics but form — and the behaviour travels with the form. Two centuries apart, two different substances, one sign change, one phenomenon. That is the series’ whole claim caught in a tabletop: the mathematics is not describing the tunnelling. The mathematics is why there is tunnelling to describe.

05 Three things tunnelling does not license

It does not borrow energy. The folklore says the particle takes out a loan from the vacuum under ΔEΔt and repays it before the universe notices. Every clause of that is wrong. The transmitted particle emerges with exactly the energy it carried in — for a wave packet, the barrier acts as a filter that slightly favours the packet’s higher-energy components, but no component changes energy and nothing outside the incident energy range ever appears. And the invoked uncertainty relation is not what it pretends to be: time is not an observable in quantum mechanics — Pauli proved there is no self-adjoint time operator conjugate to a well-behaved energy — so ΔEΔt is not a relation between two measured spreads at all. Its rigorous descendant, Mandelstam and Tamm’s 1945 inequality, bounds how fast a system’s observables can change, not what may be borrowed. Nothing is borrowed because — section 03 — nothing was ever missing: no measurement shows a deficit for the loan to cover.

It does not beat light. This one deserves care, because the temptation is real and old. MacColl noticed in 1932 that the transmitted packet shows “no appreciable delay in the transmission of the packet through the barrier”; Hartman sharpened it in 1962: the group delay saturates as the barrier thickens, so a naive traversal speed grows without bound. But the group delay tracks the peak of a drastically reshaped packet, and a peak is not a signal: the wavefront — the earliest anything can arrive — never exceeds c, and that much is consensus. What the saturated delay means is still argued. Winful’s reading — that it is the lifetime of energy stored in the barrier, a cavity time rather than a travel time — is the most influential position, and his own paper calls itself a proposed resolution. Meanwhile the experiments read different clocks and get different answers: in atomic hydrogen, attosecond streaking bounds any tunnelling delay at 1.8 attoseconds — consistent with none at all — while Steinberg’s group, timing ultracold atoms through an optical barrier with a Larmor spin clock, measured 0.61 milliseconds spent inside. These do not contradict each other; they answer different operational questions. What no one has, outside Bohmian mechanics — where trajectories exist by construction and the crossing time is definite, given the guidance equation — is a single, interpretation-neutral duration. “How long was it in there” presupposes a trajectory that neither the standard formalism nor Everett’s supplies. The question is open in the precise sense that there are several well-posed versions of it and no fact singling one out.

It does not mean you could walk through a wall. Popular treatments print a probability — ten to the minus some tower of digits — and the number is not merely small; it is unearned. The single-particle formula does not apply to a person even in principle: you are a warm, decohering, continuously monitored many-body system, not one coherent wavefunction approaching one barrier, so the calculation’s premise fails before its arithmetic starts. All the honest sentence can say is that the exponent scales with mass and width so violently that the number has no name — and that the modelling choices behind any named figure are arbitrary at the level of the exponent’s own exponent. Strike the miracle register from the other side too: tunnelling is not nature suspending law. It is law — probabilistic, energy-conserving, exact enough to build on. The flash memory holding your phone’s photographs writes its bits by Fowler–Nordheim tunnelling. A miracle you can purchase by the terabyte is neither a miracle nor an argument for one.

The particle does not defeat the wall. The wall was a statement about the classical description — and the description, not the world, is what fails inside.

06 The direction of explanation

The history is worth thirty seconds, because it undercuts the mystique from a direction philosophers will appreciate. The first tunnelling calculation was not about escape at all: Friedrich Hund, 1927, computing how a molecule flips between two mirror-image shapes — a double well, not a prison break. Within a year the applications cascaded: Oppenheimer on hydrogen ionised by a field; Fowler and Nordheim on electrons pulled cold out of metals; and — submitted within days of each other, entirely independently — Gamow, and Gurney with Condon, on alpha decay. The name arrived only afterwards; the earliest printed use historians cite is Schottky’s wellenmechanischer Tunneleffekt in 1931. Then the century industrialised it: Esaki’s tunnel diode in 1957, Josephson’s junction predicted in 1962 by a 22-year-old graduate student and confirmed within a year by Anderson and Rowell, the scanning tunnelling microscope in 1981. Whatever tunnelling is metaphysically, it is also a catalogue part. It has been shipping in consumer electronics since before the transistor radio died.

Now the inversion this series exists to perform. The popular story runs: particles are strange; strange things can defy the rules; sometimes one passes through a wall. Every clause faces backwards. Start instead from the structure and read outward. A linear equation ties curvature to a coefficient. Where the coefficient is negative the solutions turn; where it is positive they die; the crossing between the two regimes is a zero of a smooth function, not a rampart. From that single fact: the turning region is the part of the world classical mechanics could parametrise (it is exactly where a real momentum exists to parametrise it with); the dying region is the part it had to declare closed; and the faint surviving tail is the measure of how premature the declaration was. The wall does not explain the tunnelling. The sign change explains both the wall and the crossing.

Say “forbidden” carefully, then. Classical mechanics demands a real momentum at every point of a path; where V > E no real value exists, so the classical description has no solution there — a description that fails, not a territory that ends. Quantum mechanics, for its part, never assigned a sharp position and momentum simultaneously anywhere — that refusal is global, not a local emergency. What distinguishes the barrier’s interior is only that the classical pretence stops being even approximately maintainable there: outside the wall you can squint and see a marble; inside there is nothing marble-shaped to see. But the state does not blink. It persists, exponential rather than oscillatory, turning in time while dying along space, exactly as the structure said it must.

On the reading this series has been arguing — Tegmark’s, descended from Wigner’s puzzle about why mathematics fits the world so unreasonably well — none of this is a surprise to be explained away. If the physical world is a mathematical structure, then the marble, the bowl and the barrier are positions inside one, and “classically forbidden” marks the edge of a projection of the structure — the real-momentum shadow of Part Three — not an edge of the world. The tail inside the wall is the structure continuing past where its shadow gives out. The usual hedge, posted as always: physics is compatible with this reading, and suggestive of it; it does not prove it. An object-oriented realist can accept every computed number in this lesson and hold that the structure supervenes on things — that the evanescent wave describes a propensity of stuff rather than constituting the world. Both readings survive the evidence. What survives on neither reading is the claim we started from: that the wall was ever a wall.

07 The ledger

What is load-bearing: The sign of V − E deciding between oscillation and exponential decay is a theorem about linear equations, not an interpretation. The transmission formula and its exponential are exact for the rectangular barrier and verified daily — the STM’s order of magnitude per ångström, the Geiger–Nuttall line, the tunnel diode on the bench. And the impossibility of catching the deficit is a theorem too: kinetic energy is a non-negative operator, so no measurement ever shows E − V₀ < 0.
What is convention: The word “tunnelling” itself — nothing burrows, and the name postdates the physics (earliest printed use historians cite: Schottky, 1931). The rectangular barrier is an idealisation; real barriers are handled by the WKB approximation, which keeps the exponential and complicates the exponent. And the uncertainty-relation bookkeeping in section 03 is an enforcement estimate, not a derivation — the theorem stands without it.
Where the shorthand breaks: “The equation stops turning” is true of space only: the time factor e^−iEt/ħ rotates everywhere, barrier included. At exactly E = V the solutions are affine — drift, not stillness. And “the particle is inside the wall with imaginary momentum” breaks twice: the imaginary value belongs to the exponent, never to a reading, and “inside the wall” presupposes a trajectory the formalism does not supply.
Where I would push back on myself: This whole lesson reads the wavefunction realistically — the tail persists, the structure continues. A QBist or instrumentalist reads T = 0.1015 as betting odds on a detector and loses nothing predictive. And the tunnelling-time plurality can be read against me: perhaps the lesson of five well-posed clocks with five answers is that inside-the-barrier questions were never about the world at all — in which case my “the structure persists where the picture fails” is itself a picture. I think the shared mathematics of light and matter is strong evidence for the structural reading. I cannot prove the “is.”

08 Exercises

Do the stitching Derive the transmission coefficient for the rectangular barrier by matching ψ and ψ′ at both faces. Verify T + R = 1 numerically for the parameters of Figure 02. The author got a sign wrong twice on the way to that figure; the conservation check caught it both times. Let it catch yours.
Tunnel some light Take the Helmholtz equation for light beyond the critical angle and show the transverse wavevector goes imaginary — the same sign change, with no ħ anywhere. State precisely what Schrödinger’s and Helmholtz’s equations share and what they do not. What does the shared behaviour of light and electrons tell you about where the behaviour lives?
Audit the Sun Recompute the classical verdict: thermal energy at 15.7 million K, Coulomb barrier at two femtometres, Boltzmann fraction, times 10⁵⁷ protons. Then locate the Gamow peak and the penetration factor there. Finally, say what tunnelling does not explain about the Sun’s longevity, and which interaction does.
Time the crossing, five ways Define the phase delay, the dwell time, the Larmor time, the attoclock delay, and the Bohmian crossing time. For the 2019 hydrogen result and the 2020 rubidium result, state exactly which quantity each measured — and why 1.8 attoseconds and 0.61 milliseconds answer different questions rather than contradicting each other.
Argue the other side Make the instrumentalist’s best case: the evanescent tail is bookkeeping for detector statistics; nothing is “in” the barrier; section 06’s inversion is metaphysical theatre. Then tally the cost — starting with what your reading must say about the fact that Maxwell’s classical light and Schrödinger’s electrons obey the same exponential, and whether “coincidence of form” is a phrase an instrumentalist can afford.

Sources

F. Hund, Z. Physik 43, 805 (1927) — the first tunnelling calculation: molecular double wells.
G. Gamow, Z. Physik 51, 204 (1928); R. Gurney & E. Condon, Nature 122, 439 (1928) and Phys. Rev. 33, 127 (1929) — alpha decay, independently, days apart.
R. Fowler & L. Nordheim, Proc. R. Soc. A 119, 173 (1928); J. R. Oppenheimer, Phys. Rev. 31, 66 (1928) — field emission and field ionisation.
E. Merzbacher, “The Early History of Quantum Tunneling,” Physics Today 55(8), 44 (2002) — including the Schottky 1931 naming evidence.
L. A. MacColl, Phys. Rev. 40, 621 (1932) — “no appreciable delay.”
T. E. Hartman, J. Appl. Phys. 33, 3427 (1962); H. Winful, Phys. Rep. 436, 1 (2006) — the saturating delay and its proposed resolution.
U. S. Sainadh et al., Nature 568, 75 (2019) — attoclock, atomic hydrogen, delay ≤ 1.8 as.
R. Ramos, D. Spierings, I. Racicot & A. Steinberg, Nature 583, 529 (2020) — Larmor clock, 0.61(7) ms inside the barrier.
C. Dewdney & B. Hiley, Found. Phys. 12, 27 (1982); C. R. Leavens, Solid State Commun. 74, 923 (1990) — Bohmian tunnelling trajectories and times.
L. Mandelstam & I. Tamm, J. Phys. (USSR) 9, 249 (1945) — the rigorous time–energy relation.
E. Kennard, Z. Physik 44, 326 (1927) — the preparation uncertainty relation used in section 03.
I. Newton, Opticks, Book II Part I Obs. 1–8 and Book III Query 29 (2nd ed. 1717/18; 4th ed. 1730) — frustrated total internal reflection.
H. Geiger & J. M. Nuttall, Phil. Mag. 22, 613 (1911) — the empirical law; nuclear data for Figure 04 from standard evaluated tables.
L. Esaki, Phys. Rev. 109, 603 (1958); B. Josephson, Phys. Lett. 1, 251 (1962); P. Anderson & J. Rowell, Phys. Rev. Lett. 10, 230 (1963).
G. Binnig & H. Rohrer, Nobel lecture (1986) — the order-of-magnitude-per-ångström rule.
M. Tegmark, “The Mathematical Universe,” Found. Phys. 38, 101 (2008); E. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (1960).

The Wall Is a Change of Sign