Square-Well Approximation for the Anharmonic and the Double-Well Oscillators

TL;DR

This paper introduces a new approximation scheme using square-well potentials to accurately estimate energy levels in anharmonic and double-well oscillators, demonstrating non-perturbative results with broad applicability.

Contribution

The paper applies a novel approximation scheme (NGAS) with square-well potentials to anharmonic oscillators, achieving accurate, non-perturbative energy estimates and extending the method's applicability.

Findings

LO results are within a few percent of exact energies for all coupling values

The method reproduces known analytic and scaling properties of energy

Accuracy improves with perturbative corrections in the improved perturbation theory

Abstract

A novel general approximation scheme (NGAS) proposed earlier (ref.2-3) is applied to the problem of the quartic anharmonic (QAHO) and the double-well-oscillator (QDWO) in quantum theory by choosing the infinite square-well-potential in one dimension as the input approximation. The leading order (LO) results obtained for the energy eigen-values are uniformly accurate to within a few percent of the exact results for $arbitrary$ values of the quartic coupling: $\lambda > 0$ and the level-index $n_s$. These results are shown to be non-perturbative in the LO and reproduce the known analytic and scaling properties of energy as a function of the coupling $\lambda$ and $n_s$. The LO-results are further improved in accuracy by including the perturbative-correction at the next non-trivial order of an improved perturbation theory (IPT) based upon NGAS. The method can be trivially extended to other cases of higher anharmonicity.