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.
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.
This paper presents an interpretive extension of standard quantum mechanics using the Total Wave Modified Schrödinger Equation (TWMSE), reframing quantum tunneling as a dynamic process of wave interference and stress accumulation rather than a purely probabilistic event. In standard quantum mechanics, tunneling is described by exponentially decaying wavefunctions inside classically forbidden regions, often interpreted as the wave “dying out.” This work argues instead that exponential decay represents blocked propagation and stored evanescent wave deformation. Within the TWMSE framework, continuous wave incidence leads to gradual, logarithmic accumulation of interference (or “wave stress”) until a stability threshold is reached, triggering tunneling or collapse. The paper emphasizes complementarity rather than replacement: standard quantum mechanics correctly describes the spatial shape of the wave, while TWMSE provides a physical narrative for its time-dependent behavior. Analogies such as stiff springs, capacitors, and voltage drops are used to make the interpretation accessible without altering established mathematics. Appendices clarify that the TWMSE interpretation is not a hidden-variable or Bohmian theory, connect the narrative to the formal TWMSE collapse–interference functional, and explore the speculative but testable idea that tunneling rates may be tunable through resonant observer–wave coupling. This contribution is intended as a conceptual and pedagogical bridge between standard quantum tunneling theory and wave-based collapse dynamics, and as a foundation for future quantitative and experimental investigations.
Note added 10 August 2026: The experimental interpretation in this paper has been superseded. See the revised technical account, Zenodo DOI 10.5281/zenodo.21868322, which supersedes the Sorkin parameter, second harmonic, detection time estimates, and history bias claims where those exceed what follows from the minimal written dynamics. This record remains available as part of the development history. This paper presents the most complete account to date of the Total Wave Modified Schrödinger Equation (TWMSE), a deterministic framework in which quantum wavefunction collapse arises from interference between a system wavefunction and observer fields associated with the four fundamental interactions, rather than from a probabilistic postulate. The paper is organised in two parts with a formal bridge section. Part I constructs the minimal working model demonstrating well-posed pseudo-Hermitian evolution, collapse via interference thresholds, dynamical emergence of phase ergodicity via Weyl equidistribution and KAM theory, statistical recovery of the Born rule, and a quantified deviation ΔP(x) ~ δ²|Ψ|² bounded to δ ≲ 10⁻². The bridge section introduces the Two-System Falsifiability Argument: two identically prepared two-level systems with different environmental coupling histories must produce different collapse statistics under TWMSE and identical statistics under standard quantum mechanics — the conceptually minimal statement of the theory's falsifiability. Part II derives three concrete experimental signatures: the Sorkin parameter κ ~ δ²/3 from three-slit photon interference; a second harmonic fringe component at 2Δφ in Mach-Zehnder atom interferometry, strictly absent in all linear theories; and ILL-type neutron crystal interferometry. Detection of the second harmonic requires approximately 33 minutes at current cold-atom count rates. The architectural implications of TWMSE for quantum computing are developed in Appendix D of the forthcoming book Quantum Computing Is Stalling: A Way Forward (Larry Lim Kheng Cheong, forthcoming). Here is the link to the book -https://a.co/d/0hQ0I2i0 Note: A US non-provisional patent application (Application No. 19/645,198) covering collapse-based quantum computing architectures derived from this framework was filed on 12 April 2026 prior to this upload.
Note added 10 August 2026: The experimental claims in this document have been narrowed. The second harmonic in Mach Zehnder interferometry, the nonzero Sorkin parameter, the associated detection time estimates, and the two system history bias are no longer treated as derived predictions of the minimal TWMSE dynamics. They are retained as open derivation targets. Readers should consult the revised technical account, Zenodo DOI 10.5281/zenodo.21868322, for the current status of these claims. A plain-English companion to the technical paper "From Mathematical Foundation to Experimental Test: A Unified Account of the Total Wave Modified Schrödinger Equation (TWMSE)" (April 2026). For 100 years, quantum mechanics has had no explanation for why particles collapse from a spread-out wave of possibilities into a definite dot on a detector. This document explains a new idea — the Total Wave Modified Schrödinger Equation (TWMSE) — in plain language for anyone curious about quantum physics. TWMSE proposes that collapse is not magical or random. It is the natural result of interference between the particle's own wave and background waves tied to nature's four fundamental forces. When these waves align strongly enough, the fuzzy cloud snaps into a real, localised particle. No mystery. No consciousness required. The document covers: why the Born rule emerges naturally from TWMSE rather than being assumed; the Two-System Thought Experiment showing why two identically prepared particles with different interaction histories must behave differently; the specific experiment testable today using cold-atom interferometry in existing laboratories; and what it means if the experiment finds — or does not find — the predicted signal. No equations. No prior physics knowledge required. The full mathematical treatment is in the companion technical paper. The architectural implications for quantum computing are developed in the forthcoming book Quantum Computing Is Stalling: A Way Forward (Larry Lim Kheng Cheong, 2026 - https://a.co/d/0hQ0I2i0 ).
Quantum mechanics describes phenomena that are key to understanding processes that cannot be explained by classical physics. This paper analyzes the behavior of particles in the presence of a potential barrier through numerical solutions to both time-dependent and time-independent Schrödinger equations. These results enabled a detailed understanding of the energy structure and temporal dynamics of wave functions, including the phenomenon of quantum tunneling. The obtained data have a wide range of applications in technology: from semiconductor devices to quantum computing.
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 paper presents a revised unified account of the Total Wave Modified Schrödinger Equation (TWMSE), a proposed deterministic collapse framework in which quantum state reduction is investigated as a possible dynamical consequence of interaction between a system wavefunction and observer fields associated with physical interactions. The purpose of this revision is to distinguish clearly between mathematical structures that have been explicitly formulated, consequences that follow from the minimal written dynamics, and stronger experimental proposals that remain to be derived. A central result is that a spatially homogeneous scalar collapse term can modify the global amplitude of a quantum state but cannot, by itself, generate spatial localization or branch selection. After normalization, the common scalar factor cancels from the spatial probability distribution. Likewise, a spatially homogeneous multiplicative metric can restore a conserved physical norm but does not by itself produce a measurable spatial probability bias. The paper also corrects the treatment of the time dependent metric. Conservation of the physical norm requires the full time dependent metric condition, G˙+ℏi(H†G−GH)=0, rather than the time independent pseudo Hermitian relation alone. For the minimal scalar TWMSE model, the corresponding metric compensates exactly for the non Hermitian amplitude evolution, while also revealing a long time invertibility issue when the collapse functional possesses a nonzero mean. These results change the status of several experimental signatures proposed in the earlier unified TWMSE account. In particular, a nonzero Sorkin parameter, a second harmonic in Mach Zehnder interferometry, associated detection time estimates, and an already established branch dependent history bias are not treated in this revised paper as derived predictions of the minimal homogeneous TWMSE dynamics. They are retained instead as open derivation and experimental targets that require a genuinely spatially structured, operator level, or branch sensitive probability dynamics. The two system history experiment is consequently reformulated as a falsifiability target. A completed TWMSE model would need to demonstrate mathematically how different observer field histories could produce different normalized outcome statistics for systems having the same final standard quantum state, while remaining distinguishable from ordinary Hamiltonian evolution, decoherence, systematic effects, and competing collapse models. The paper further identifies the principal mathematical requirements for completing the framework, including branch sensitive collapse dynamics, existence and consistency of the time dependent metric, derivation of the physical probability law, strong coupling phase dynamics, accumulated interference and memory, multi particle consistency, no signalling, relativistic completion, conservation laws, parameter identification, and derivation or revision of the earlier interferometric proposals. This paper represents the author's current technical position on these questions and supersedes the experimental interpretation of the earlier technical unified account, Zenodo DOI 10.5281/zenodo.19537024, where those claims exceed what follows from the minimal written dynamics. The earlier technical paper remains part of the public development record. Its plain English companion, Zenodo DOI 10.5281/zenodo.19537305, likewise remains part of that record and should be read in light of the present revision. The objective of this revised account is therefore narrower but stronger: to state explicitly what TWMSE presently derives, what remains conditional, what remains open, and what must be demonstrated before the framework can yield a distinctive, normalized, causal, and experimentally testable prediction. Revision notice: This publication represents the author's current technical position on the experimental status of TWMSE. It supersedes the interpretation of experimental claims in Zenodo DOI 10.5281/zenodo.19537024 where those claims exceed what follows from the minimal homogeneous dynamics. The plain English companion to that earlier account, Zenodo DOI 10.5281/zenodo.19537305, should likewise be read in light of this revision. Both earlier records remain publicly available as part of the development history.