---
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  "title": "OP-QM-005 · Core–Field Coupling in Tunneling",
  "title_en": "OP-QM-005 · Core–Field Coupling in Tunneling",
  "category": "Quantum Phenomena",
  "summary": "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.",
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    "current_state": "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."
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    "conditions": [
      "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."
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---
## Established 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.

<Equation name="qm-transmission" />

## Research 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.

## The 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.

## Scope 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](/en/theory/quantum/tunneling) · [accompanying field](/en/theory/quantum/accompanying-field) · [detection statistics](/en/theory/quantum/detection-statistics) · [tunneling](/en/theory/quantum/tunneling) · [accompanying field](/en/theory/quantum/accompanying-field) · [detection statistics](/en/theory/quantum/detection-statistics) · [quantum](/en/theory/quantum)

<CanonicalSource />