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OP-QM-005 · Core–Field Coupling in Tunneling

For a one-dimensional stationary rectangular barrier with identical propagation parameters on both sides, the exact envelope boundary solution is established: the hyperbolic transmission form equals its exponential rational form, and reflection plus transmission equals one. A variable-coefficient barrier has a divergence-form equation, amplitude transport and first-order WKB expression. These solve specified effective-envelope propagation problems.

Based on the final manuscript · 第五章

TheoryTheory · OPENReviewMathematical review · INCOMPLETEEvidenceEvidence · UNTESTEDCanonicalNoStatus, review and sources
OP-QM-005 · P2

NEEDS THEORY · Needs theory

Quantum foundationsNumerical physics
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01Established results

For a one-dimensional stationary rectangular barrier with identical propagation parameters on both sides, the exact envelope boundary solution is established: the hyperbolic transmission form equals its exponential rational form, and reflection plus transmission equals one. A variable-coefficient barrier has a divergence-form equation, amplitude transport and first-order WKB expression. These solve specified effective-envelope propagation problems.

ASD-WEB-QM-031
T=[1+V02sinh⁡2(κa)4E(V0−E)]−1,z=e−2κa,D=V02(1−z)2+16E(V0−E)z,T=16E(V0−E)zD,Rrefl=V02(1−z)2D,T+Rrefl=1,jII=2ℏκm0Im⁡(AB∗)\begin{aligned}T&=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1},\\ z&=e^{-2\kappa a},\quad D=V_0^2(1-z)^2+16E(V_0-E)z,\\ T&=\frac{16E(V_0-E)z}{D},\quad R_{\rm refl}=\frac{V_0^2(1-z)^2}{D},\quad T+R_{\rm refl}=1,\\ j_{\rm II}&=\frac{2\hbar\kappa}{m_0}\operatorname{Im}(AB^*)\end{aligned}

Mathematical result
Exact boundary solution for equal exterior parameters, real potential and 0<E<V₀. z,D abbreviate 5.3.34; Rrefl is reflectance, not core radius. Complex exponential superposition carries flux.
Chapter reference: Ch. 5 (5.3.32)–(5.3.35) · ASD-BOOK-ddbade997b45

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02Research still required

Evolve the finite core, accompanying field, barrier material and back reaction from one action. Track topology and energy exchange and compare core-crossing probability with the envelope flux ratio. Return to the second-order hyperbolic equation for rapidly changing boundaries; retain WKB slow-variation conditions and turning-point connections for variable barriers.

03The next reproducible deliverable

Deliver a joint initial-boundary evolution for a specified barrier, including conservation residuals, grid/boundary convergence, the model basis of core trajectories and statistical initial conditions. Reproduce the exact rectangular result and threshold limit before reporting departures from the effective description.

04Scope and research links

An equality defining a WKB expression does not make the true transmission of every barrier exact; nonzero evanescent-envelope flux alone does not prove nonlinear-core crossing.

tunneling · accompanying field · detection statistics · tunneling · accompanying field · detection statistics · quantum

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STATUS & REVIEW
Theory StatusTheory · OPEN
Mathematical ReviewMathematical review · INCOMPLETE
Overall Evidence StatusEvidence · UNTESTEDNOT YET UNIQUELY TESTED · known wave phenomena are requirements; an independent ASD source and universal event statistics remain open.
Canonical ASDNo · Research notesCanonical research framework · effective limits are conditional
Module version 1.0.0Reviewed 2026-09-16ASD-BOOK-ddbade997b45Metadata reviewed 2026-09-22
Adopted source, conditions and previous versions

第五章 §3.7—§3.9 式（5.3.28）—（5.3.39）

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  • An equality defining a WKB expression does not make the true transmission of every barrier exact; nonzero evanescent-envelope flux alone does not prove nonlinear-core crossing.

Source locators: p1983 · p1988 · p1993 · p1995 · p1998 · p2000 · p2008 · p2009

Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e

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QM-0accdcbe7725 · Ch. 5 §§3.7–3.9

CH5-ca737ec4e327 · 第五章-出版整理稿.docx
ca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262

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第五章 · 第五章 §3.7—§3.9 式（5.3.28）—（5.3.39）

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CITATION & SOURCE FILE

ASD: OP-QM-005 · Core–Field Coupling in Tunneling. OP-QM-005-EN. Version 1.0.0; 2026-10-24. https://asd-physics.org/en/open-problems/quantum/tunneling-core-coupling. 第五章 §3.7—§3.9 式（5.3.28）—（5.3.39）. Source ASD-BOOK-ddbade997b45.

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