New Developments in Time-Independent Quantum-Mechanical Perturbation Theory

TL;DR

This paper introduces two innovative perturbation techniques for solving the time-independent Schrödinger equation, improving convergence and applicability, especially for degenerate systems, demonstrated through harmonic oscillator examples.

Contribution

It presents a matrix-based iterative solution and a synthetic Hamiltonian approach, enhancing perturbation methods beyond standard series expansions.

Findings

Iterative method outperforms standard perturbation series.

Synthetic Hamiltonian improves convergence for degenerate states.

Methods successfully applied to various harmonic oscillator problems.

Abstract

This report discusses two new ideas for using perturbation methods to solve the time-independent Schr\"odinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schr\"odinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.