Quantum Tunneling

Tunneling is not a particle passing through a wall.
It is an evanescent envelope penetrating a region where
oscillatory propagation is suppressed — carrying the possibility,
never a fraction, of one complete matter excitation.

Objective

Quantum tunneling is one of the clearest places where the classical particle picture fails. In classical mechanics, a localized object with energy E cannot enter a region where the potential energy exceeds E. Yet quantum systems are observed to cross classically forbidden barriers: alpha decay, field emission, tunnel diodes, Josephson junctions, and scanning tunneling microscopy all depend on the effect, and tunneling has been implicated in selected electron- and proton-transfer processes in chemistry and biology.

The standard treatment is mathematically complete at its own level: the wavefunction does not vanish inside the barrier but decays exponentially, and a finite barrier leaves nonzero amplitude on the far side. What the standard treatment leaves open — deliberately, in its instrumental form — is the physical picture: what is the entity doing in the barrier, how is energy accounted for, and what do the competing “tunneling time” definitions measure? HGUT proposes a mechanical layer beneath the formalism, and this chapter develops it with the discipline the newer volumes require: the mathematical reduction is presented with its approximation stated, the ontological claims are separated from the derived ones, and the open seams are named.

Three interpretive puzzles frame the chapter. The energetic puzzle: how does anything occupy a region where it “lacks the energy”? The trajectorial puzzle: where is the particle inside the barrier, and how long does crossing take? The ontological puzzle: what crosses? The HGUT answers, in brief: no energy is borrowed, because E < V is a statement about the local dispersion relation, not about an energy deficit; no point-particle trajectory crosses, because what penetrates is the evanescent quantum envelope; and what is transferred, when transmission succeeds, is one complete, indivisible matter excitation — never a fraction.

The purpose of this chapter is to derive the leading tunneling law

from the HGUT complex phase field equation, to interpret each ingredient mechanically, and to keep the ontology consistent with the modern matter sector — where an electron or an alpha particle is a stable topological Harrison-Knot configuration, not a free phase ripple.

The Ontological Frame: Identity vs. Envelope

One distinction, sharpened since this chapter was first drafted, governs everything that follows:

In modern HGUT, an electron is a stable, charged, topological Harrison-Knot configuration; an alpha particle is an organized four-nucleon matter state. That topological identity is not erased during propagation, and this chapter must not erase it either. What is distributed — and what becomes evanescent inside a barrier — is the effective center-of-mass quantum envelope associated with one conserved matter excitation: the field-level description of where the complete excitation may be found and with what weight.

For matter tunneling, the modern statement is: one matter occupation/topological sector is conserved throughout,

its effective center-of-mass envelope is distributed; that envelope becomes evanescent in the barrier; and a successful transmission event places thecomplete matter excitation in the outgoing region. Tunneling does not require a classical point-particle trajectory through the forbidden region, and it does not leave a fractional electron or fractional alpha particle inside the barrier. This is Integer B of the measurement chapters, applied to matter: the envelope divides; the occupation quantum does not.

How the medium maintains and relocates the complete topological configuration when its envelope amplitude spans a barrier — the microscopic story connecting the knot’s structural identity to the envelope’s evanescent support — is the chapter’s deepest open item, and it is flagged as such rather than papered over:

[OPEN — microscopic continuity of topological identity during tunneling]

The Collective-Coordinate Assumption

The distinction between topological identity and distributed envelope creates a further reduction that must be stated explicitly. A stable Harrison-Knot matter configuration is an extended nonlinear field structure. The scalar envelope ψ(x,t) used in the tunneling calculation is therefore not automatically identical to the complete microscopic knot field.

This chapter assumes an effective collective-coordinate regime in which the internal topological configuration remains in one stable state while its center coordinate X is described by an envelope Ψ(X,t). Schematically,

with internal excitations, deformation channels, fragmentation, and multi-channel conversion neglected.

Under this rigid-state, one-channel approximation, the center-of-mass envelope may obey the effective Schrödinger equation derived below. Deriving that collective-coordinate reduction from the full nonlinear Harrison-Knot dynamics remains open.

[OPEN — collective-coordinate reduction from topological matter to the effective one-body envelope]

Physical Setup

Consider a one-dimensional barrier of width a occupying 0 < x < a. Outside the barrier, the medium is in its ordinary propagation regime. Inside, the effective envelope equation acquires a local spectral-gap or detuning term. At the level of the phase-sector field equation this is modeled as an increase in the effective stiffness term:

One caution attaches to this modeling step, and it changes the status of an identification the earlier draft treated as universal. Real barriers — nuclear Coulomb barriers, semiconductor band gaps, vacuum work functions, Josephson junctions, molecular barriers — arise from different microscopic physics. HGUT proposes that in each case the effective envelope-level term V(x) descends from changes in the local collective response of the medium — potentially involving stiffness, charge circulation, bonding, density, or phase coupling. The system-specific reduction from the underlying medium variables to V(x) has not been performed. The stiffness form above is therefore a [CANDIDATE] realization of the barrier, not a derived universal identity; what the derivation below actually requires is only that the envelope equation contain a local term raising the propagation threshold. [OPEN — system-specific derivation of V(x)].

The incident excitation has envelope energy E < V. The HGUT question is not how a particle crosses the barrier. It is:

The answer is: it becomes evanescent.

HGUT Phase Field Equation

The complex phase field obeys the second-order medium equation

with χ the inertial coefficient of the phase sector, α the phase stiffness, M²(x) the local effective gap term, and φ the full complex oscillating field. In the homogeneous vacuum M² = M₀² and the carrier frequency is

The tunneling calculation studies what happens when M²(x) is locally raised above M₀².

Carrier–Envelope Decomposition and the Schrödinger Limit

Write the field as a slowly varying envelope on the carrier, φ(x,t) = ψ(x,t) e^−iω₀t, and for stationary scattering take ψ(x,t) = ψ(x) e^{−iω_env t}, so the full frequency is ω = ω₀ + ω_env. Substituting φ = ψ(x)e^−iωt into the phase field equation:

Splitting M² = M₀² + δM²(x) and expanding

the vacuum carrier terms cancel via M₀² = χω₀², leaving

Multiplying by ℏ/(2χω₀) and using the identifications of the complex-phase-field chapter,

yields the stationary Schrödinger equation

The status of this reduction must be stated exactly. The Schrödinger equation is not introduced independently here; it is obtained as the positive-frequency, slowly varying, low-energy envelope limit of the candidate HGUT phase equation. Its validity is conditional on the scale separation

which is what licenses dropping ω_env² against 2ω₀ω_env in the frequency expansion above; for time-dependent envelopes it additionally requires the slow-envelope approximation suppressing ∂²ψ/∂t² against ω₀ ∂ψ/∂t. [CANDIDATE — envelope reduction; approximation stated], pending closure of the master field equation itself.

The Barrier Region

Inside a rectangular barrier, V(x) = V₀ > E for 0 < x < a, and the Schrödinger equation rearranges to

The general barrier solution is

Both terms matter, and the earlier draft’s dismissal of the growing exponential must be corrected. For a semi-infinite barrier, regularity as x → ∞ selects the decaying exponential alone. For afinite barrier bounded at x = 0 and x = a, both exponentials generally appear, with complex coefficients C and D fixed by continuity of ψ and dψ/dx at the two interfaces. The full matched combination, not the decaying term in isolation, is the physical barrier solution — and, as shown below, it is the full combination that carries the transmitted current.

Mechanical Interpretation of the Evanescent Mode

In an ordinary propagation region, the envelope supports oscillatory solutions ψ ~ e^ikx: propagating phase modes the medium can carry forward while preserving phase rotation. Inside the barrier, the local dispersion relation changes character. The cleanest statement of what E < V means — and the replacement for all “insufficient energy” and “borrowed energy” talk — is:

The conserved energy eigenvalue simply does not admit a real wavenumber in that region. The oscillatory mode continues analytically to imaginary momentum: e^ikx → e^−κx. The field does not vanish at the barrier boundary; it penetrates as a non-propagating deformation of the phase medium.

This is familiar from ordinary wave physics — evanescent electromagnetic fields beyond total internal reflection, acoustic penetration of impedance barriers — and HGUT identifies quantum tunneling as the same class of phenomenon in the complex phase field of the vacuum medium, with one addition the classical analogies lack: the occupation structure introduced in “The Ontological Frame: Identity vs. Envelope,” which is what makes the detected outcome a whole quantum rather than a partial wave.

Transmission Through a Finite Barrier

The transmission coefficient must be defined as a current ratio, not an endpoint-amplitude ratio — a correction to the earlier draft that matters both of principle and of practice. With the probability current

the transmission coefficient is

Two facts sharpen the evanescent picture rather than damaging it. First, a single decaying exponential ψ = Ce^−κx with fixed overall phase carries zero stationary current, j = 0: an isolated evanescent tail transports nothing. Second, the nonzero stationary tunneling current through a finite barrier lives in the full matched combination Ce^−κx + De^+κx, whose complex coefficients encode the boundary conditions and permit a conserved nonzero j. For

the barrier current is

Thus neither exponential is independently a traveling mode; the conserved current is encoded in their boundary-fixed relative phase. The barrier region contains an evanescent field configuration, and transmission is encoded by the complete boundary-matched solution, not by treating one decaying exponential as a propagating wave.

For a real, time-independent rectangular barrier with V = 0 in the two asymptotic regions, constant effective mass m, no absorption or inelastic channels, and identical media on the two sides, boundary matching gives the exact result

If the effective gradient coefficient, inertia, or mass changes across an interface, the matching law must be re-derived from the underlying conserved flux. Continuity of ψ and dψ/dx is specific to the constant- coefficient model used here; a variable-coefficient reduction generally requires continuity of the appropriate flux-weighted derivative, such as α dψ/dx in the underlying phase equation or (1/m) dψ/dx in an effective-mass formulation.

In the thick-barrier limit κa ≫ 1,

The accurate leading statement is therefore

with the endpoint-amplitude ratio |ψ(a)|²/|ψ(0)|² ~ e^−2κa supplying the correct exponent and the boundary matching supplying the prefactor.

The exponential WKB structure of this law appears across alpha decay, field emission, STM, semiconductor tunneling, and related systems — with system-specific barrier profiles, prefactors, many-body corrections, and coupling laws in each case; the rectangular formula is the pedagogical skeleton, not the quantitative description of every system at once. (The Josephson effect in particular involves coherent pair tunneling and macroscopic phase dynamics, not merely the one-particle rectangular formula.) The tunneling rate decreases exponentially with barrier width a, with the gap V₀ − E, and with the effective mass m. On the last of these, the earlier draft’s reading was too narrow: from the mapping above, m = ℏχω₀/α can be large because of greater inertial loading χ, a higher carrier scale ω₀, or reduced gradient stiffness α — a combination of medium properties, not “stiffness” alone.

Causality: How the Barrier Field Is Established

One sentence of the earlier draft must be removed outright: the evanescent field is notestablished “essentially instantaneously across the barrier.” A stationary scattering solution describes the asymptotic response after the boundary-value problem has settled; it does not describe the causal formation of that state. The low-energy Schrödinger envelope equation is not itself suitable for determining a front velocity. Causal propagation must be read from the full second-order medium equation (the phase field equation above), whose characteristic structure governs the buildup, reshaping, storage, and release of the barrier field.

Accordingly, no superluminal conclusion may be drawn from the stationary envelope solution or from the saturation of a particular group-delay definition. HGUT must demonstrate causal front propagation in the complete time-dependent theory and connect that characteristic cone to the observable signal cone. [OPEN — causal time-dependent tunneling dynamics].

Tunneling Time: Reframed, Not Dissolved

If a particle traverses a barrier, how long does it take? The literature offers several non-equivalent definitions — phase time, dwell time, Büttiker–Landauer time, Larmor time — which can disagree by orders of magnitude for the same process, and attosecond-scale experiments have not produced consensus on “the” tunneling time.

The earlier draft claimed the question is dissolved because “there is no single crossing time.” The modern position is more careful, in both directions. HGUT does reject one thing: a classical point-particle transit time as the fundamental description — there is no localized particle tracing a path through the barrier whose stopwatch reading the definitions compete to report. But rejecting the point-particle transit time does not make the measured timescales unreal. Phase time, dwell time, and Larmor-type times answer different operationalquestions — wave packet delay, barrier occupancy, response of a co-measured degree of freedom — and their disagreement shows that “tunneling time” is not one uniquely defined observable without a specified measurement protocol, not that the observables are artifacts. The attosecond experiments measure real physical timescales; HGUT’s obligation is to derive those operational times from the time-dependent collective equations of the medium, protocol by protocol.

The stationary calculation of this chapter cannot close that question, and it is entered as open rather than declared dissolved. [OPEN — operational tunneling times from causal dynamics].

What Actually Crosses the Barrier

The common phrase “the particle tunnels through the barrier” is misleading, but the correction must respect the matter ontology. The sequence is:

1. A matter excitation — one conserved topological identity with a distributed center-of-mass envelope — approaches the barrier.
2. The barrier region admits no oscillatory envelope mode at the incident energy (the dispersion statement above).
3. The envelope becomes evanescent inside the barrier; the boundary-matched configuration connects the two sides.
4. The matched solution carries a small conserved transmitted current (the transmission-coefficient definition above).
5. The scattering dynamics establish reflected and transmitted outgoing weights. In an observed experiment, environmental or detector coupling converts those alternatives into a realized whole-quantum outcome through the first-passage architecture. If transmission is realized, the complete matter occupation appears in the outgoing sector; no fractional remnant remains in the barrier or reflected sector.

Energy Conservation

Tunneling is sometimes described as the particle temporarily “borrowing energy” to cross. HGUT needs no such language, and the precise statement is cleaner than the earlier draft’s.

Conservation for a scattering state is expressed locally, through the continuity equation, and globally, through flux balance — not through the total integrated energy of an infinitely extended stationary state, which is not a finite localized quantity:

with the appropriate sign convention for the reflected flux. Nothing is created or destroyed; the incident current divides between reflection and transmission.

And the “energy deficit” framing is answered by the dispersion statement, not by an energy loan: E < V does not mean the state must temporarily acquire energy V; it means the conserved eigenvalue E admits no real wavenumber in the barrier region — k² < 0, k = iκ. The classical implication “E < V ⇒ no particle trajectory” is correct and remains correct; what replaces it is “E < V ⇒ evanescent envelope.” There is no violation of energy conservation. There is a failure of the particle-trajectory description.

Alpha Decay

Alpha decay is the canonical example, and it is where the topological firewall matters most. An alpha-like cluster is confined by the nuclear well and separated from the exterior by the Coulomb barrier. In HGUT language: the effective center-of-mass amplitude of the preformed alpha cluster is evanescent through the Coulomb barrier. A successful outgoing event preserves the alpha cluster’s completecharge, mass, and occupation identity — what escapes is a whole alpha particle, never a fractional one, and never a bare phase ripple that “reconstructs” an alpha particle from nothing.

The escape rate is controlled by the WKB attenuation factor

whose exponential sensitivity to the barrier profile explains the enormous range of alpha lifetimes and underlies the Geiger–Nuttall systematics (with cluster-preformation factors and nuclear-structure corrections supplying the system-specific prefactors). HGUT reproduces this structure because it reproduces the evanescent-envelope equation; what it adds is the identity discipline, and what it owes is the microscopic account of how the cluster’s topological configuration remains whole while its envelope spans the barrier — the same open continuity item flagged above.

Scanning Tunneling Microscopy

STM exploits the same law: a conducting tip is held angstroms from a surface, and the current depends exponentially on the gap,

with κ ~ 1 Å⁻¹ for typical work functions — so a 0.1 Å change in distance changes the current by roughly 20%, which is what makes atomic-resolution imaging possible.

The HGUT reading, with the firewall in place: the electronic state has evanescent support across the vacuum gap, producing a transition amplitude between tip and sample states. The transferred electron remains an indivisible charged topological excitation; the distributed tunneling amplitude is not a fractional electron smeared through the gap. The vacuum gap is not an absence — it is the medium in a configuration that suppresses ordinary electron propagation while supporting evanescent envelope coupling — and the STM image maps the strength of that coupling, which is sensitive to the local electronic structure of the surface.

Detection: The Measurement Architecture Beyond the Barrier

The tunneling calculation determines the reflected and transmitted field weights. Those weights are properties of the scattering dynamics and do not depend on a later laboratory detector. A detector samples the outgoing state; it does not create the transmission coefficient. When an outgoing alternative is registered, that registration is described by the measurement architecture locked in the Wave–Particle Duality chapter, and the older “basin-selection” language of earlier drafts is retired here in its favor.

The transmitted envelope determines the relative far-side exposure,

at leading order. The measurement architecture then converts that distributed exposure into a localized event through first-passage ignition: exposed sites accumulate toward threshold from run-varying microscopic readiness; the first to ignite claims the one available matter occupation quantum; depletion lockout prevents a second completed event, and no fractional remnant is left in the barrier or the reflected arm. The detection probability inherits the exponential suppression because the first-passage rates are weighted by the transmitted exposure — and that rate law, r_i ∝ |ψ_i|², remains the shared open seam of the measurement program; it is used here, not derived here. [OPEN — exposure-to-rate reduction (shared Seam 1)].

The tunneling probability therefore combines two facts of different standing: the evanescent decay of the envelope through the barrier ([CANDIDATE — within the envelope reduction], exact given the Schrödinger limit) and the exposure-weighted single-quantum detection architecture ([CLOSED — HGUT architectural level], rate law open). No new probability postulate enters at the barrier.

What This Chapter Has Established

• The stationary Schrödinger equation is obtained as the positive-frequency, slowly varying, low-energy envelope limit of the HGUT phase field equation, valid for |ω_env| ≪ ω₀. [CANDIDATE — envelope reduction; approximation stated].
• In a region with V > E the envelope equation admits evanescent solutions with κ = √(2m(V−E))/ℏ; the barrier solution is the boundary-matched combination of both exponentials, and a lone decaying exponential carries zero current. [ESTABLISHED — within the reduction].
• Transmission is the current ratio T = j_trans/ j_inc; for the rectangular barrier T = [1 + V₀²sinh²(κa)/4E(V₀−E)]⁻¹, with thick-barrier form T ≈ (16E(V₀−E)/V₀²) e^−2κa — hence T ∝ e^−2κa. [ESTABLISHED — within the reduction].
• E < V is a dispersion statement (k² < 0), not an energy deficit; no energy is borrowed, and conservation is carried by the continuity equation and flux balance. [ESTABLISHED].
• The matter ontology is respected throughout: one conserved topological identity, one occupation quantum, a distributed envelope that becomes evanescent, a whole quantum transferred on success. [CANDIDATE — HGUT ontological frame].
• The use of a scalar center-of-mass envelope assumes a rigid-state, one-channel collective-coordinate reduction of the nonlinear topological matter configuration. [OPEN — collective-coordinate derivation].
• Detection beyond the barrier proceeds by the locked first-passage architecture, inheriting the exponential suppression through exposure-weighted rates. [CLOSED — HGUT architectural level; rate law open].

What This Chapter Has Not Established

Status Ledger

Summary

Quantum tunneling is not a violation of energy conservation, not a particle borrowing energy, and not a particle passing through a wall. It is the ordinary behavior of a continuous phase envelope in a region whose local dispersion relation forbids oscillation but permits evanescent penetration — attached, in HGUT, to a matter ontology in which the excitation whose envelope penetrates is one conserved topological identity, transferred whole or not at all.

The law T ∝ e^−2κa is not a mystery and not a postulate: it is the current ratio of a boundary-matched evanescent solution, with the exponent set by the decaying envelope and the prefactor by the interface matching.

The next chapter turns to the Hong–Ou–Mandel effect, where the same occupation discipline meets two-quantum interference: two indistinguishable excitations at a beam splitter, and the deletion of the coincidence sector by interference of the both-transmitted and both-reflected alternatives.