Non-Relativistic Quantum Mechanics in Multidimensional Geometric Frameworks

TL;DR

This paper develops a generalized non-relativistic quantum mechanics framework within multidimensional geometric spaces characterized by power-law dispersion relations, extending the Schrödinger equation and spectral analysis to higher-order geometries.

Contribution

It introduces a novel geometric formulation of quantum mechanics based on higher-order spatial derivatives and dispersion relations, expanding the traditional quadratic kinetic energy model.

Findings

Spectral energies scale as rac{(2n+1)^j}{

Eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms.

The uncertainty principle remains valid in the generalized framework.

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 Schr\"odinger 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.