Ground and excited states of spherically symmetric potentials through an imaginary-time evolution method: Application to spiked harmonic oscillators

TL;DR

This paper introduces an imaginary-time evolution method combined with energy minimization and orthogonality constraints to accurately compute ground and excited states of 3D central potentials, demonstrated on harmonic oscillators and Morse potentials.

Contribution

The paper presents a novel finite-difference imaginary-time evolution approach for solving 3D Schrödinger equations, capable of efficiently obtaining high-quality wave functions and energies for various potentials.

Findings

Accurate wave functions and energies for 3D harmonic oscillator.

Competitive results for Morse and spiked harmonic oscillators.

Method outperforms some existing techniques in accuracy.

Abstract

Starting from a time-dependent Schr\"odinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literature data reveals that this is able to produce high-quality and competitive results. The method could be useful for this and other similar potentials of interest in quantum mechanics. Future and outlook of the method is briefly discussed.