This repository presents a two-part exploration of negative transit times in quantum tunneling, combining a pedagogical explanation rooted in standard physics with a speculative extension inspired by the Unified Applicable Time (UAT) and Unified Causal Principle (UCP) frameworks. Part 1 — Path-Integral Analysis and Numerical Simulation:We explain the experimentally observed negative group delay of photons traversing an ultracold atomic cloud via the principle of least action in Feynman's path-integral formulation. The dominant contribution comes from non-classical trajectories that avoid the energetic cost of the barrier. A numerical simulation of a Gaussian wave packet tunneling through a rectangular potential barrier reproduces the negative transit time, confirming the interpretation within standard quantum mechanics. Part 2 — The Hartman Effect and the Causal Coherence Limit:We review the Hartman effect —the saturation of the tunneling (phase) time to a finite, negative value— and provide an analytical calculation for an electron incident on a rectangular barrier. The phase time saturates at a fraction of a femtosecond, regardless of barrier width, guaranteeing that local temporal fluctuations never accumulate to violate macroscopic causality. We then offer a speculative interpretation through the lens of the UAT/UCP frameworks (DOIs: 10.5281/zenodo.17729221, 10.5281/zenodo.17718670), suggesting that the saturation is a manifestation of a deeper causal coherence limit that protects the global causal structure. Contents of the repository: LaTeX source and compiled PDF of the unified article. Python script for the numerical wave-packet simulation (Part 1). Python script for the analytical phase-time calculation (Part 2). Generated figures. Note: This work is a curiosity-driven intellectual exercise. The UAT/UCP connection is presented as a speculative perspective and does not form part of mainstream physics. The standard quantum mechanical analysis, however, is rigorous and self-contained. Related DOIs: UAT: 10.5281/zenodo.17729221 UPC: 10.5281/zenodo.17718670 Antifrequency: 10.5281/zenodo.18809178 Author: Miguel Ángel Percudani (Puan, Buenos Aires, Argentina)License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Description (Unified): This repository presents a two-part exploration of negative transit times in quantum tunneling, combining a pedagogical explanation rooted in standard physics with a speculative extension inspired by the Unified Applicable Time (UAT) and Unified Causal Principle (UCP) frameworks. Part 1 — Path-Integral Analysis and Numerical Simulation: We explain the experimentally observed negative group delay of photons traversing an ultracold atomic cloud via the principle of least action in Feynman's path-integral formulation. The dominant contribution comes from non-classical trajectories that avoid the energetic cost of the barrier. A numerical simulation of a Gaussian wave packet tunneling through a rectangular potential barrier reproduces the negative transit time, confirming the interpretation within standard quantum mechanics. Part 2 — The Hartman Effect and the Causal Coherence Limit: We review the Hartman effect —the saturation of the tunneling (phase) time to a finite, negative value— and provide an analytical calculation for an electron incident on a rectangular barrier. The phase time saturates at a fraction of a femtosecond, regardless of barrier width, guaranteeing that local temporal fluctuations never accumulate to violate macroscopic causality. We then offer a speculative interpretation through the lens of the UAT/UCP frameworks (DOIs: 10.5281/zenodo.17729221, 10.5281/zenodo.17718670), suggesting that the saturation is a manifestation of a deeper causal coherence limit that protects the global causal structure. Contents of the repository: LaTeX source and compiled PDF of the unified article. Python script for the numerical wave-packet simulation (Part 1). Python script for the analytical phase-time calculation (Part 2). Generated figures. Note: This work is a curiosity-driven intellectual exercise. The UAT/UCP connection is presented as a speculative perspective and does not form part of mainstream physics. The standard quantum mechanical analysis, however, is rigorous and self-contained. Related DOIs: UAT: 10.5281/zenodo.17729221 UPC: 10.5281/zenodo.17718670 Antifrequency: 10.5281/zenodo.18809178 Author: Miguel Ángel Percudani (Puan, Buenos Aires, Argentina) License: Creative Commons Attribution 4.0 International (CC BY 4.0)
This submission presents a conceptual framework that interprets several open problems in quantum mechanics through the lens of the Unified Applicable Time (UAT) and Unified Causal Principle (UCP) frameworks. It does not claim to replace quantum mechanics; it offers a complementary geometric and thermodynamic layer of interpretation. The document identifies three manifestations of a single principle —irreducible fluctuation— at three scales: cosmological (7% thermal calibration margin), laboratory (force metric R_geom = 0.2791 from the 8-coil interferometer), and quantum (~0.07% detector resolution limit). This principle provides interpretive bridges to: The measurement problem (collapse as saturation of the Ivancho limit, κ_crit = 4.978). Decoherence (information leakage governed by the irreducible fluctuation margin). Entanglement (geometric projection from higher-dimensional causal structures). The quantum vacuum (vacuum energy density predicted by the UAT cosmological constant derivation). Superposition (reinterpreted via the Causal Oscillation Hypothesis as a detector resolution artifact, not a real physical state). Negative tunneling time (Hartman effect explained by least-action paths in the Feynman path integral, with UAT/UCP providing the causal consistency condition). Limitations are explicitly acknowledged: UAT/UCP does not derive the Schrödinger equation, the Born rule, or second quantization. Whether the framework can be extended to these domains is identified as an open question for future investigation. The package includes the LaTeX manuscript. All referenced UAT/UCP works are publicly available on Zenodo with their respective DOIs.
This submission presents a conceptual framework that interprets several open problems in quantum mechanics through the lens of the Unified Applicable Time (UAT) and Unified Causal Principle (UCP) frameworks. It does not claim to replace quantum mechanics; it offers a complementary geometric and thermodynamic layer of interpretation. The document identifies three manifestations of a single principle —irreducible fluctuation— at three scales: cosmological (7% thermal calibration margin), laboratory (force metric R_geom = 0.2791 from the 8-coil interferometer), and quantum (~0.07% detector resolution limit). This principle provides interpretive bridges to: The measurement problem (collapse as saturation of the Ivancho limit, κ_crit = 4.978). Decoherence (information leakage governed by the irreducible fluctuation margin). Entanglement (geometric projection from higher-dimensional causal structures). The quantum vacuum (vacuum energy density predicted by the UAT cosmological constant derivation). Superposition (reinterpreted via the Causal Oscillation Hypothesis as a detector resolution artifact, not a real physical state). Negative tunneling time (Hartman effect explained by least-action paths in the Feynman path integral, with UAT/UCP providing the causal consistency condition). Limitations are explicitly acknowledged: UAT/UCP does not derive the Schrödinger equation, the Born rule, or second quantization. Whether the framework can be extended to these domains is identified as an open question for future investigation. The package includes the LaTeX manuscript. All referenced UAT/UCP works are publicly available on Zenodo with their respective DOIs.
Note: This paper is the technical companion to the interpretive study “Beyond Exponential Decay: Interpreting Quantum Tunneling as Wave Stress Accumulation” (Zenodo DOI: 10.5281/zenodo.18001177). Standard quantum tunneling is commonly described using stationary solutions of the Schrödinger equation, in which wavefunctions decay exponentially inside classically forbidden regions. In a recent interpretive work, tunneling was reexamined within the Total Wave Modified Schrödinger Equation (TWMSE) as a process of wave stress accumulation, in which exponential spatial decay coexists with slow, logarithmic-like temporal buildup under continuous incidence (Larry Lim Kheng Cheong, 2025, Zenodo, https://doi.org/10.5281/zenodo.18001177). That study was intentionally presented in a conceptual and pedagogical manner to make the physical interpretation accessible to a broad audience. The present paper serves as a direct technical continuation of that interpretive study. Its purpose is to provide an explicit time-dependent formulation, numerical setup, and illustrative simulation of tunneling within the TWMSE framework, without diluting the interpretive clarity of the earlier work. By separating interpretation from formal development, the two papers together address distinct audiences: the first emphasizing physical intuition, and the present work focusing on mathematical structure and dynamical behavior. A minimal collapse–interference functional is introduced for a one-dimensional finite potential barrier, and its effect under continuous wave incidence is examined. Numerical simulations demonstrate that while the spatial wave profile inside the barrier retains its standard exponential decay, the accumulated interference exhibits sublinear, logarithmic-like temporal saturation over time due to feedback in the modified dynamics. Standard quantum mechanics is recovered as a limiting case. These results provide a concrete, testable illustration of how time-dependent accumulation and threshold-like behavior can emerge in tunneling systems while preserving full compatibility with established quantum mechanics.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.
The time a quantum particle spends traversing a classically forbidden region is among the most persistently contested observables in quantum theory, its definition splintering across the Wigner phase time, the Büttiker–Landauer clock, and complex-time path-integral constructions. The debate has centered almost exclusively on a single number—the mean delay—and on the Hartman effect, whereby that mean saturates to a width-independent constant for an opaque barrier, suggesting an unbounded and apparently superluminal traversal speed. Yet an attoclock measures not a number but a distribution: the observable imprinted on the photoelectron spectrum is a temporal spread, not a delay. The narrow-band limit of this variance is fixed by the time-energy uncertainty relation alone,
Var(τ)→ℏ2/4ΔE2, a general property of any finite-bandwidth wave packet. What has not been shown is that this floor survives intact through a barrier deep in the Hartman-opaque regime, where the mean delay has already saturated and where naive intuition expects the pulse to broaden. We prove, for the rectangular barrier, that the transmitted variance remains locked onto this incident-packet value independently of the barrier width
d—a spectral-pinning theorem. Equivalently, the dimensionless product
Var(τ)ΔE2 collapses onto a single universal plateau shared by all bandwidths, a data collapse confirmed numerically to better than 0.5% across two decades of opacity and verified against the predicted inverse-square bandwidth scaling
Varnarrow/Varbroad=(ΔEbroad/ΔEnarrow)2 to within a few percent, the small residual excess (28.56 versus 25) being itself a quantitative signature of barrier-induced reshaping. The genuine new result is the finite-bandwidth correction responsible for this excess: the opaque barrier acts as a coherent spectral filter—a tunneling analogue of the gain-narrowing mechanism familiar from ultrafast optics—that reweights the transmitted band and causes the variance to decrease weakly with opacity [log-log exponent
α≈+0.070 in the convention
Var∝(κd)−α], while the mean arrival time is pushed above the Hartman plateau. We derive the leading filter coefficient
λ2 in closed form, verify it against an independent, exactly solvable smooth barrier (Eckart), and connect both the pinned floor and its filtering correction to experiment through the attoclock map
σθ≃ωστ. The result recasts the tunneling-time question from the fragile arithmetic of a saturating mean to the robust, universal statistics of a spectrally pinned—and quantifiably filtered—width.