New exact solutions of the 3D Schr\"odinger equation

TL;DR

This paper presents new exact solutions to the 3D Schrödinger equation for a quadratic funnel potential, including explicit energy spectra, eigenfunctions, and electromagnetic solutions with vortex structures, analyzed via the Wigner-Vlasov formalism.

Contribution

It introduces a novel class of exact solutions for the 3D Schrödinger equation with a quadratic funnel potential, incorporating gauge invariance and vortex phenomena.

Findings

Explicit energy spectrum and eigenfunctions derived.

Electromagnetic solutions with Dirac string magnetic fields obtained.

Superpositions reveal diverse vortex and probability current fields.

Abstract

Previously we found a unique quantum system with a positive gauge-invariant Weyl-Stratonovich quasi-probability density function which can be defined by the so-called {\guillemotleft}quadratic funnel{\guillemotright} potential [Phys. Rev. A 110 02222 (2024)]. In this work we have constructed a class of exact solutions to the 3D Schr\"odinger equation for a two-parameter {\guillemotleft}quadratic funnel{\guillemotright} potential based on the -model of micro and macro systems. Explicit expressions for the energy spectrum and the set of eigenfunctions have been found. Using gauge invariance for scalar and vector potentials, a solution to the electromagnetic Schr\"odinger equation has been obtained, with a magnetic field in the form of a {\guillemotleft}Dirac string{\guillemotright} defined by a singular vortex probability flux field. Superpositions of eigenfunctions leading to various types of vortex and potential probability current fields have been investigated in detail. The analysis of the quantum system's properties has been carried out within the Wigner-Vlasov formalism.