Angular invariant quantum mechanics in arbitrary dimension

TL;DR

This paper generalizes fundamental one-dimensional quantum mechanics problems to arbitrary dimensions with radial symmetry, facilitating visualization and connections to advanced theories like string theory.

Contribution

It introduces a method to extend classic quantum problems to higher dimensions using angular invariance, linking simple models to complex multi-dimensional physics.

Findings

Solutions involve Bessel and Whittaker functions

Provides a framework for visualizing multi-dimensional quantum states

Connects quantum mechanics to string theory concepts

Abstract

One dimensional quantum mechanics problems, namely the infinite potential well, the harmonic oscillator, the free particle, the Dirac delta potential, the finite well and the finite barrier are generalized for finite arbitrary dimension in a radially symmetric, or angular invariant, manner. This generalization enables the Schr\"{o}dinger equation solutions to be visualized for Bessel functions and Whittaker functions, and it also enables connections to multi-dimensional physics theories, like string theory.