Eigenstates with the auxiliary field method

TL;DR

This paper demonstrates how the auxiliary field method can be used to derive accurate approximate eigenstates for two-body Schrödinger equations with various potentials, extending its application to specific potential forms.

Contribution

It introduces analytical approximations for eigenstates of two-body Schrödinger equations with linear, logarithmic, and exponential potentials using the auxiliary field method.

Findings

High accuracy of eigenstate approximations for certain potentials

Extension of auxiliary field method to specific potential forms

Analytical expressions for eigenstates with characteristic sizes

Abstract

The auxiliary field method is a powerful technique to obtain approximate closed-form energy formulas for eigenequations in quantum mechanics. Very good results can be obtained for Schr\"odinger and semirelativistic Hamiltonians with various potentials, even in the case of many-body problems. This method can also provide approximate eigenstates in terms of well known wavefunctions, for instance harmonic oscillator or hydrogen-like states, but with a characteristic size which depends on quantum numbers. In this paper, we consider two-body Schr\"odinger equations with linear, logarithmic and exponential potentials and show that analytical approximations of the corresponding eigenstates can be obtained with the auxiliary field method, with a very good accuracy in some cases.