Energy-Moment Methods in Quantum Mechanics

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Abstract
Three quantum-mechanical computational techniques based on energy moments, μk= ∫ dqψ*(q)Hkψ(q), and semimoments, 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.

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