Tensorial Formulation of the Quantum Harmonic Oscillator in Curvilinear Coordinates

Resumen

This article develops a tensorial framework for the analysis of the threedimensional isotropic quantum harmonic oscillator in curvilinear coordinates. Starting from the metric tensor and the associated geometric structures, we derive the line element, scale factors, surface and volume elements, as well as the Beltrami–Laplacian operator in orthogonal systems. Within this geometric setting, the stationary Schrödinger equation is solved in Cartesian, cylindrical, and spherical coordinates. The separation of variables naturally leads to families of orthogonal polynomials, Hermite and associated Laguerre; whose orthogonality is dictated by the corresponding Riemannian measures. The resulting spectrum, EN = ℏω (N + 32), exhibits the characteristic degeneracy gN = (N+1)(N+2)/2 , reflecting the isotropy of the potential. Regularity and self-adjointness conditions are discussed, ensuring the physical validity of the eigenfunctions. The metric and Laplacian in cylindrical elliptic coordinates are also introduced, laying groundwork for future studies on separability and anisotropy.