Non-Relativistic Quantum Mechanics in Multidimensional Geometric Frameworks

• URL: http://arxiv.org/abs/2603.26826v1
• Date: Fri, 27 Mar 2026 01:27:51 GMT
• Title: Non-Relativistic Quantum Mechanics in Multidimensional Geometric Frameworks
• Abstract summary: formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks.<n>The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries.<n>The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity.
• Score: 0.32116198597240847
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation \(E \propto |p|^{j}\), where \(j = N - 1\). Starting from the generalized Minkowski distance in \(L^j\)-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent \(j\)-th order Schrödinger equation. The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries. While plane-wave solutions and translational invariance are preserved, the spectral structure is modified, with bound-state energies scaling as \((2n+1)^{j}\), leading to cubic and quartic growth in higher geometries. The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity. A generalized probability framework based on \(j\)-fold conjugation is introduced, ensuring a real-valued probability density and consistent expectation values. Despite these generalizations, the Heisenberg uncertainty principle is preserved. The formulation presents quantum mechanics as a geometry-dependent theory in which dispersion relations, spectral properties, and probabilistic structure emerge from the underlying spatial metric.

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