SRMs meet Schrödinger: Multiple-ratio expansions for quantum dynamics and entanglement in a timeless dialogue.
Description
If Schrödinger had had a ruler in 1926 that was already multiscale, he would probably have derived discretizations that reconcile phase coherence and nonlocality with fewer points. Infinite Series with Multiple Ratios (SRMs) provide such a ruler: a scheme with multiple coupling steps that preserves unitarity and introduces controlled homologous connections - a fertile ground for studying entanglement as a structural regularity under uncertainty. We propose SRM-Schr, a multiscale scheme for the time-dependent Schrödinger Equation that replaces the classical Laplacian with a SRM Laplacian (stencils with multiple ratios) and introduces a homologous coupling operator that induces calibratable nonlocal correlations. The evolution is done by unitary split-step (Crank–Nicolson/Strang), ensuring conservation of probability. The choice of SRM ratios is guided by physical scaling relations. (E = hf, λ = h/p), to align resonance and phase coherence without "forcing" determinism: we maintain the probabilistic nature but organize the multiscale structure of the wavefield. In 1D tests (free particle, oscillator, double barrier), SRM-Schr reduces dispersion and phase error compared to uniform discretizations at the same cost. In two-body models, an SRM "pulse" smoothly controls entanglement (von Neumann entropy). Results suggest that SRMs form a natural basis for simulating interference and entanglement with fewer points and greater fidelity.
Technical info (En)
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8 – Citation CAP 10 – LIVRO FUCAP NO ZENODO
França, C. R. (2025). Método de criptografia Heru Technologies: único do mundo que utiliza fórmulas matemáticas inéditas autorais e que criptografa com perturbações binárias. Zenodo. https://doi.org/10.5281/zenodo.15653585
9 - França, C. R. (2025, junho 23). Heru Technologies encryption method: unique in the world that uses unpublished mathematical formulas and encrypts with binary disturbances. https://doi.org/10.5281/zenodo.15717868
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13 - França, C. R. (2025). SRMs and the Middle Path: Bridging Relativity and Quantum Mechanics. Zenodo. https://doi.org/10.5281/zenodo.17069526
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