Optimized Perturbation Theory for Wave Functions of Quantum Systems

TL;DR

This paper extends optimized perturbation techniques from energy calculations to wave functions in quantum systems, enabling accurate, asymptotically correct solutions even under strong coupling conditions.

Contribution

It introduces a method to construct wave functions using an envelope of perturbative solutions, generalizing the optimized perturbation approach.

Findings

Produces uniformly valid wave functions with correct asymptotics

Effective even for strong coupling regimes

Applicable to quantum anharmonic oscillator and double well potential

Abstract

The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings.