Energy‐Moment Methods in Quantum Mechanics

Abstract

Three quantum‐mechanical computational techniques based on energy moments, μk = ∫ dqψ*(q)Hkψ(q)μk=∫dqψ*(q)Hkψ(q), and semimoments, νk(q′) = [Hkψ(q)]q = q′νk(q′)=[Hkψ(q)]q=q′, are formulated. The μ method, which employs the μk, is connected to the method of moments in probability theory, to the variational method, and to eigenvalue spectroscopy. The ν and λ methods, which employ semimoments, are related to local energy methods using one and several configuration points, respectively. An Nth‐order calculation, requiring 2N moments or semimoments, yields N approximate eigenvalues and eigenfunctions. In accordance with a conjectured convergence criterion, exact eigenstates are approached in the limit N→∞. From quantities obtained in a moments calculation, a lower bound on the ground‐state eigenvalue can also be determined using a refinement of Weinstein's criterion. A computational method for generating moments and semimoments is given and the μ method is applied to the linear harmonic oscillator.