Quantum tunnelling weighed by the wave itself

Does probability really leak through a barrier taller than the particle's energy at exactly the rate scattering theory claims — and can the transmission coefficient, Gamow's exponential decay law, and the Ramsauer–Townsend resonance all be read off a numerically evolved wavefunction that was never told any of those formulas?

Units: transmission probability T at E = 2.88, V₀ = 3.2, a = 1.6 (κ = 0.8 exactly, κa = 1.28)

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The finding

Quantum tunnelling weighed by the wave itself: a Crank–Nicolson Schrödinger evolution with ONLY the barrier potential coded (no sinh² formula anywhere in the dynamics) returns T = 0.1156335 vs the closed-form 0.1156337 (Richardson rel 1.1e-6, grid-invariant to 2.8e-3 over 24 meshes) — Gamow's exponential wall d(lnT)/da = −1.60249 vs −2κ = −1.6, the Ramsauer–Townsend transparency resonance at k₂a = π to 2.1e-3, and the classical particle falsified on BOTH sides: it tunnels where Newton forbids (T_cl = 0) and reflects where Newton guarantees passage (T_cl = 1, wave gives 0.408)

Method

Evolve i∂ψ/∂t = −½ψ'' + V(x)ψ (ħ = m = 1) with the exactly-unitary Crank–Nicolson scheme — deliberately a DIFFERENT integrator from the module's Visscher leapfrog — for a Gaussian packet (k₀ = 2.4, E = 2.88) hitting a rectangular barrier (V₀ = 3.2, a = 1.6; the module's own defaults). Read the per-momentum transmission as the spectral ratio T(k) = |ψ_transmitted(k)|²/|φ₀(k)|², dividing only where the incident spectrum holds ≥15% of its max (the Young lesson). Score against the textbook T = [1+V₀²sinh²(κa)/(4E(V₀−E))]⁻¹, which appears ONLY in the scoring path. Gates: Richardson dx-extrapolation with a measured 2nd-order convergence ratio (3.99×); 24 jittered discretizations; whole-band T(k) tracking; machine-precision unitarity (4.7e-14 over 3000 steps); a barrier-width sweep recovering Gamow's slope −2κ; the k₂a = π transparency resonance located from the curve; and the two-sided classical falsification.

Measurements, controls & cross-checks

Grid invariance

Configs

24

Mean

0.115414

Se

1.6200e-5

Worst rel

0.00279

Note

jitter over dx ∈ {0.05, 0.04, 0.032}, packet width s ∈ {4..7}, launch x₀, dt ∈ {0.008..0.0125}; residual is pure O(dx²) Laplacian dispersion (δlnT ≈ 2a·δκ, ≈1.4% at dx = 0.05), removed by Richardson: convergence ratio 3.99 on dx-halving, extrapolated rel 1.1e-6

Band tracking

K points

111

Worst rel

0.000738

Note

one run yields the whole T(k) curve across E = 2.2 → 3.6, crossing V₀; every point sits on the two-branch (sinh²/sin²) law

Gamow slope

Recovered

-1.60249

Expected

-1.6

Rel error

0.00156

Note

lnT vs a over a ∈ [2.4, 3.6] while T falls 25×; each width also individually on the law to ≤1.7e-2. The exponential wall that makes alpha-decay half-lives span 24 decades (Gamow 1928)

Ramsauer townsend

Recovered k2

1.96753

Known k2

1.9634954

Rel error

0.00205

T at resonance

1

Note

the barrier turns perfectly transparent exactly at k₂a = π, located as the vertex of 1/T − 1 from a packet launched above the barrier — never coded

Unitarity drift

4.7300e-14

Classical rival

Below

E < V₀: classical T = 0 exactly; the wave delivers T = 0.1156 — probability crosses a wall the particle cannot climb

Above

E = 3.682 > V₀ at k₂a = π/2: classical T = 1 exactly; the wave reflects 59% (T = 0.40813 vs QM law 0.40937, rel 3.0e-3) — wrong in kind on both sides, not merely in magnitude

What it reduces to

The rectangular-barrier transmission coefficient of stationary scattering theory (Griffiths §2.6; Gamow 1928 for the thick-barrier exponential; Ramsauer 1921 / Townsend for the transparency resonance), recovered the way Goldberg, Schey & Schwartz (1967) first watched it happen — by evolving a wavepacket — but scored quantitatively through the spectral ratio |ψ_T(k)|²/|φ₀(k)|². Non-circular in the same sense as the young/blackbody/hydrogen oracles: the evolution code contains only the Hamiltonian (kinetic term + a rectangular V), never the sinh²/sin² law, never κ, never the resonance condition; Crank–Nicolson is also a different integrator from the module's Visscher scheme, so module and oracle agree through physics, not shared code. NEW ESTIMATOR LESSON (#5, the time-domain sibling of Young's denominator rule): a ratio observable is also biased by an undrained TRANSIENT — at t = 30 the wide-barrier (a = 3.6) evanescent interior still held ~1.2e-3 of probability, the size of the entire transmitted signal, faking T +13% and a Gamow slope of −1.52; measuring at t = 40 (remnant < 2% of T) restored −1.6025. Gate the remnant-to-signal ratio, and fix the clock, never the tolerance.

Module systematics

The module's on-screen 'transmitted T' integrates the total probability beyond the barrier for its σ = 5 packet — a momentum-spread AVERAGE of T(k), not the plane-wave T(k₀) shown as its 'theory' line; the two numbers legitimately differ and share no code path (the theory label is _theoryT(), called only from render's HUD string). The systematics are now MEASURED BY EXECUTION, not estimated: the executed screen shows T = 0.091027 at the plateau vs the oracle's CN packet average ⟨T⟩ = 0.136577 — a net −33.4% screen bias that decomposes into the module's coarse dx = 0.2 grid/full-height-barrier-edge systematic (−34.2%, first-order in dx: the effective width ≈ a + dx documented earlier is real — the float64 twin family dx = 0.2/0.1/0.05 converges with ratio 2.36, first order, NOT the oracle's second order, because the module fills ALL cells j ∈ [jL, jR] at full V₀ with no half-height edges) stacked against the +18.2% momentum-spread bias, plus a float32-vs-float64 gap of only 4.1e-8. Carried to the continuum by first-order Richardson, the module's own discretization meets the oracle: E1 = 0.138283 vs ⟨T⟩ = 0.136577, gap 1.71e-3 ≤ the DERIVED ceiling 3.95e-3 (= |E1−E2| twin tail 3.88e-3 + CN tail 7.3e-5 + dt tail 1.6e-7 — every term measured, none fitted). An earlier version of this note ESTIMATED the edge bias at 'roughly −15%' from e^(−2κ·dx); execution shows the combined grid effect is −34.2% — the estimate was off 2×, which is exactly why disclosed systematics must be measured, not guessed. CERTIFIED (module-honesty gates M–Q, zero module edits): the derisk EXECUTES the shipped QuantumTunnelModule.ts (sha256-pinned, 28 exact strip pairs, new Function with Babylon/DOM stubs) — 800 fl(1/120) engine calls of the real fixedUpdate/render are bit-exact vs a statics-built float32 Visscher replica at EVERY call (full R/I/prob fields, T/R/norm/acc, thin-instance counts + every active buffer byte, the %5 HUD write cadence and text), the display is LIVE (the packet relaunches at the replica-predicted engine calls 310/620, final HUD ≠ first, final HUD sha-pinned), the executed _theoryT() === the oracle's scoring lawT BIT-FOR-BIT (identical source text held to ===, 4.6e-13 from the reference constant = its own 12-digit print rounding), the plateau HUD shows the measured 9.1% beside the theory 11.6% — two numbers, two code paths — sha-pinned, and the executed measurement path (_launch/_step/_computeProb/_writeField/fixedUpdate) is answer-free (no sinh, no 0.1156…, no _theoryT call; _theoryT referenced exactly twice — definition + render's label; scanner proven live by a planted violation). Tamper self-tests: known→0.1256 fails A+B+C+D(+O's known-gap check) with the recovered value byte-unchanged; cert sha flip fails ONLY M; T_screen pin flipped by one ulp fails ONLY O (execution gates not vacuous).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- tunnel (scripts/tunnel-derisk.mjs)

Oracle

scripts/oracles/tunnel.reference.json

Sources

G. Gamow, Z. Phys. 51, 204 (1928). C. Ramsauer, Ann. Phys. 64, 513 (1921); J. S. Townsend & V. A. Bailey (1922). D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §2.6. A. Goldberg, H. M. Schey & J. L. Schwartz, Am. J. Phys. 35, 177 (1967) — the classic computer-generated wavepacket-on-a-barrier pictures. J. Crank & P. Nicolson, Proc. Camb. Phil. Soc. 43, 50 (1947). P. B. Visscher, Comput. Phys. 5, 596 (1991) — the module's scheme.