Extended Power Method to Calculate Pre-Selectable Eigenvalues and Eigenstates

TL;DR

This paper introduces an extended power method for quantum systems that enables pre-selection of eigenvalues and eigenstates of commuting operators, improving the calculation of excited states and handling degeneracies.

Contribution

The paper reformulates the power method for quantum operators, allowing pre-selection of eigenvalues and eigenstates, which enhances the calculation of excited states and degeneracies.

Findings

Method can calculate excited states.

Handles degeneracy effectively.

Accelerates convergence in eigenvalue problems.

Abstract

The quantum mechanical expression relating two commuting operators is reformulated such that the power method (also called method of moments) for iteratively calculating eigenvalues and eigenvectors becomes applicable. The new iterative scheme thus obtained allows to pre-select a quantum number of one of the two commuting operators and then calculate a corresponding eigenvalue of the other operator. The result is the common eigenvector of the eigenvalue pair. Among others the method may be used to calculate excited states, cope with degeneracy and/or accelerate convergence. Small example calculations are presented as a 'Proof-of-Concept' and to reveal some properties of the new method.