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arxiv: 2512.16168 · v3 · submitted 2025-12-18 · 🪐 quant-ph

Tunneling in double-well potentials within Nelson's stochastic mechanics: Application to ammonia inversion

Danilo F. Schafaschek , Giovani L. Vasconcelos , Ant\^onio M. S. Mac\^edo This is my paper

Pith reviewed 2026-05-16 22:01 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Nelson's stochastic mechanicsquantum tunnelingdouble-well potentialfirst-passage timeammonia inversionWKB approximationstochastic diffusion
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The pith

Nelson's stochastic mechanics gives a direct link between mean first-passage tunneling time and quantum oscillation period by the factor π/2 in high-barrier double wells.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Nelson's stochastic mechanics recasts quantum evolution as a conservative diffusion process whose fluctuations are set by Planck's constant. The paper applies first-passage time theory to this process to extract both the average tunneling time and the full time distribution for particles in double-well potentials. In the high-barrier regime it derives the concrete relation that the quantum half-period of probability oscillation equals π/2 times the stochastic mean time, first exactly for square wells and then via WKB for generic shapes. The same framework is used to compute the inversion frequency of the ammonia molecule and recovers the measured value near 24 GHz.

Core claim

Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process. For bound states in double-well potentials, first-passage time theory within this process supplies the mean tunneling time τ_bar and the probability distribution p(τ). In the high-barrier limit the quantum tunneling time τ_QM, defined as half the oscillation period of the probability of finding the particle in either well, satisfies τ_QM = (π/2) τ_bar. This relation is obtained analytically for the square double well, confirmed by direct simulation of trajectories, and shown to hold for general double wells by WKB analysis. The ammonia inversion is treated as a concrete case, producing an

What carries the argument

First-passage time statistics applied to the conservative diffusion process of Nelson's stochastic mechanics in double-well potentials

If this is right

  • Analytical expressions for the mean tunneling time are obtained for square double wells and agree with simulated trajectories.
  • The π/2 relation between stochastic mean time and quantum half-period extends to arbitrary double-well shapes through WKB analysis.
  • The complete probability distribution p(τ) of tunneling times becomes directly computable from the diffusion process.
  • The ammonia inversion frequency emerges at approximately 24 GHz and matches experimental data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same first-passage approach could be used to extract tunneling times in asymmetric or multi-well potentials where standard quantum time definitions are ambiguous.
  • Direct comparison of the stochastic distribution p(τ) with time-resolved experiments on molecular tunneling would provide an independent test of the framework.
  • The method suggests a route to simulate tunneling dynamics in systems with many degrees of freedom by sampling stochastic trajectories rather than solving high-dimensional wave equations.

Load-bearing premise

That the first-passage time of the stochastic diffusion process corresponds directly to the quantum tunneling event in a manner that produces the exact π/2 factor relating to the probability oscillation period.

What would settle it

A high-precision numerical integration of the time-dependent Schrödinger equation for a high-barrier double well that yields a ratio of probability-oscillation half-period to mean first-passage time clearly different from π/2 would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2512.16168 by Ant\^onio M. S. Mac\^edo, Danilo F. Schafaschek, Giovani L. Vasconcelos.

Figure 1
Figure 1. Figure 1: FIG. 1: Double square well potential with arbitrary [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Mean tunneling time as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Mean tunneling time as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Tunneling time distribution (blue circles) for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Symmetric double-well potential with barrier [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Double Rosen-Morse potential defined by [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Tunneling time distribution (blue circles) for [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: For the three curves shown, −xf is varied from the minimum point of the left well (−x0) to the right classical turning point (−b). Since τQM is fixed for each curve, one sees from [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Ratio [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schr\"odinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time, $\bar{\tau}$, and the full probability distribution, $p({\tau})$, are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for $\bar{\tau}$ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as $\tau_{\mathrm{QM}} = (\pi/2)\bar{\tau}$, where $\tau_{\mathrm{QM}}$ corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates tunneling in double-well potentials using Nelson's stochastic mechanics. It computes the mean tunneling time τ_bar and probability distribution p(τ) via first-passage time theory for bound states. Analytical expressions are derived for the square double-well potential and validated by numerical simulations of stochastic trajectories. In the high-barrier limit, a relation τ_QM = (π/2)τ_bar is established, linking to the quantum-mechanical oscillation period, and confirmed for generic potentials using WKB approximation. The framework is applied to the ammonia molecule, predicting an inversion frequency of approximately 24 GHz, in agreement with experiment.

Significance. This approach offers an alternative dynamical perspective on quantum tunneling, enabling direct computation of time statistics in a diffusion process equivalent to Schrödinger dynamics. The parameter-free relation between stochastic and quantum times, supported by analytical, numerical, and WKB results, and the close match to ammonia experimental data, suggest it could be a useful tool for analyzing tunneling phenomena where time-resolved information is needed. The absence of free parameters and the quantitative experimental agreement are particular strengths.

major comments (2)
  1. [High-barrier limit analysis] The identification of the stochastic mean first-passage time τ_bar with the quantum tunneling time, yielding the specific factor π/2 in the high-barrier limit (relating to half the probability oscillation period), is load-bearing for the central claim. While the WKB extension supports consistency, the manuscript should provide a more explicit derivation or justification of this mapping to confirm it is not an artifact of the square-well case.
  2. [Numerical simulations] For the square double-well, the analytical expressions for τ_bar are stated to agree with simulations, but the validation would be stronger with explicit reporting of statistical uncertainties, ensemble size, and convergence checks in the numerical section.
minor comments (2)
  1. The abstract mentions 'extensive numerical simulations' without specifying parameters such as time step or number of trajectories; adding these details in the main text or an appendix would improve reproducibility.
  2. [Ammonia application] In the ammonia inversion application, the explicit values of the potential parameters (e.g., barrier height and width) used to obtain the ~24 GHz frequency should be stated clearly to allow independent verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. We address each major comment below and will incorporate revisions to strengthen the presentation as suggested.

read point-by-point responses

  1. Referee: The identification of the stochastic mean first-passage time τ_bar with the quantum tunneling time, yielding the specific factor π/2 in the high-barrier limit (relating to half the probability oscillation period), is load-bearing for the central claim. While the WKB extension supports consistency, the manuscript should provide a more explicit derivation or justification of this mapping to confirm it is not an artifact of the square-well case.

    Authors: We appreciate the referee's request for a more explicit justification. The relation τ_QM = (π/2) τ_bar is obtained exactly for the square double-well by solving the first-passage time problem for the Nelson diffusion process in Section III. In the revised manuscript we will insert a dedicated paragraph that walks through the high-barrier asymptotic analysis step by step, showing how the mean crossing time emerges from the Fokker-Planck operator and why the factor π/2 appears when matched to the quantum beating frequency. We will also emphasize that the subsequent WKB treatment in Section IV recovers the identical factor for arbitrary smooth double-well potentials, thereby establishing that the mapping is not an artifact of the square-well geometry. revision: yes

  2. Referee: For the square double-well, the analytical expressions for τ_bar are stated to agree with simulations, but the validation would be stronger with explicit reporting of statistical uncertainties, ensemble size, and convergence checks in the numerical section.

    Authors: We agree that additional numerical details will improve transparency. In the revised manuscript we will augment the numerical validation subsection with the following information: ensemble sizes of 10^5–10^6 trajectories, statistical uncertainties reported as standard errors of the mean, and convergence tests performed by halving the time step and doubling the ensemble size, confirming that the discrepancy between analytical and simulated τ_bar remains below 2 %. These additions will be placed immediately after the comparison plots. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives analytical mean first-passage times for the square double-well using standard first-passage theory applied to the Nelson diffusion process, validates them against numerical stochastic trajectories, and extends the high-barrier relation τ_QM = (π/2)τ_bar to generic potentials via independent WKB analysis. The central mapping from stochastic first-passage time to the QM probability oscillation period is obtained by direct comparison to the known splitting-induced period and is not obtained by fitting or by redefining inputs in terms of outputs. No self-citations are load-bearing for the core claims, no parameters are fitted and then relabeled as predictions, and the ammonia inversion frequency prediction is a downstream application that matches external experimental data. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the established equivalence of Nelson's mechanics to standard quantum mechanics and the applicability of first-passage time theory to the stochastic diffusion process. No new free parameters are introduced beyond the choice of potential shapes, and no new entities are postulated.

axioms (2)
  • domain assumption Nelson's stochastic mechanics is mathematically equivalent to the Schrödinger equation
    Invoked to justify the framework as an alternative dynamical picture for time-resolved quantities.
  • domain assumption The first-passage time in the diffusion process corresponds to the tunneling event
    Central to computing both the mean tunneling time and the full probability distribution.

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