Auxiliary fields as a tool for computing analytical solutions of the Schr\"{o}dinger equation

TL;DR

This paper introduces an auxiliary field method to derive approximate analytical solutions for the Schrödinger equation with arbitrary bound-state potentials, enhancing understanding of energy spectra in quantum systems.

Contribution

It presents a novel auxiliary field technique that yields accurate analytical solutions for complex potentials, improving upon existing formulas in quantum mechanics.

Findings

Provides analytical solutions for power-law and logarithmic potentials.

Offers highly accurate energy formulas for specific potentials.

Enhances qualitative analysis of bound state spectra.

Abstract

We propose a new method to obtain approximate solutions for the Schr\"{o}dinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows in many cases to find analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which improve a lot the corresponding formulae that can be found in literature.