Quantization of the charge in Coulomb plus harmonic potential

TL;DR

This paper investigates the quantization of charge and energy in systems described by a Schrödinger equation with combined Coulomb and harmonic potentials, revealing that both are quantized and depend on quantum states, with two quantum numbers needed.

Contribution

It demonstrates that charge and energy quantization occur in these models and highlights the necessity of two quantum numbers for systems involving Heun's equation.

Findings

Charge and energy are quantized and state-dependent.

Two quantum numbers are required to describe radial degrees of freedom.

The results suggest a general feature of differential equations with higher singularities.

Abstract

We consider two models where the wave equation can be reduced to the effective Schr\"odinger equation whose potential contains both harmonic and the Coulomb terms, $\omega^{2}r^{2}-a/r$. The equation reduces to the biconfluent Heun's equation, and we find that the charge as well as the energy must be quantized and state dependent. We also find that two quantum numbers are necessary to count radial degrees of freedom and suggest that this is a general feature of differential equation with higher singularity like the Heun's equation.