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International Journal of Quantum Chemistry

Computation of expectation values with Dirichlet one‐dimensional wave functions

Marco A. Núñez

Published Jan 5, 1995• 8 citations
Publisher

Abstract

Abstract Recently, it was proven that the wave functions ψ of one‐dimensional Schrödinger operators can be approximated by those ψn corresponding to the Dirichlet eigenproblem in a finite box sufficiently large. In this article, sufficient conditions to compute expectation values are used to prove that the expectation value 〈ψ, Sψ〉 can be approximated by 〈ψn, Sψn〉 as n → ∞, for a wide class of unbounded symmetric operators S. © 1995 John Wiley & Sons, Inc.

DOI
10.1002/qua.560530105

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Abstract Recently, it was proven that the wave functions ψ of one‐dimensional Schrödinger operators can be approximated by those ψn corresponding to the Dirichlet eigenproblem in a finite box sufficiently large. In this article, sufficient conditions to compute expectation values are used to prove that the expectation value 〈ψ, Sψ〉 can be approximated by 〈ψn, Sψn〉 as n → ∞, for a wide class of unbounded symmetric operators S. © 1995 John Wiley & Sons, Inc.
The eigenvalue problem in of Schrödinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls. It is shown that the Dirichlet and the von Neumann problems in generate upper and lower bounds, respectively, to the eigenvalues of the unbounded system. To illustrate the method, rapidly converging two-sided bounds for the energy levels of two- and three-well oscillators are presented, using simple trigonometric basis functions.
It has been recently proven that the bound states of the one-dimensional Schr\"odinger equation H${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$=E(i){\mathit{E}}^{(\mathit{i})}E(i)${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$ in [0,\ensuremath{\infty}) can be approximated by those of the corresponding Dirichlet eigenproblem HR{\mathit{H}}_{\mathit{R}}HR​${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$=ER(i){\mathit{E}}_{\mathit{R}}^{(\mathit{i})}ER(i)​${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$ in a finite box [0,R] when R\ensuremath{\rightarrow}\ensuremath{\infty}. In this paper, we use a set of mathematical criteria that guarantee the correct calculation of expectation values in the frame of Hilbert spaces, to show that correct expectation values of most physical operators S are obtained by using the equation ${\mathrm{lim}}_{\mathit{R}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{j})}$〉= 〈${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}^{(\mathit{j})}$〉, which includes operators not relatively form bounded by the Hamiltonian H. It is shown that standard numerical methods supply approximate wave functions ${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$ for which the equation ${\mathrm{lim}}_{\mathit{m}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{j})}$〉= 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{j})}$〉 holds true. Thus, combining the previous limits we obtain a general approach to compute correct expectation values. It is shown how this approach agrees with the results found by other authors in special cases. In the particular case of computing approximate wave functions ${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$ with the Ritz method, we show, analytically and numerically, that correct convergence toward the expectation values of high power of moment operators rk{\mathit{r}}^{\mathit{k}}rk holds even when the basis functions have an analytic structure that substantially differs from that of the exact wave function.
Abstract An explicitly connected commutator expansion for the average value of an observable in the coupled‐cluster theory is derived. Specifically, it is shown that the expectation value of an operator for the state Ψ, related to the Fermi vacuum Φ by the exponential Ansatz ψ = eT Φ, is expressed as a finite commutator series containing the cluster operator T and an auxiliary operator S, defined by a linear equation involving again a finite commutator series in T. The above result is applied to derive the explicitly connected commutator form of the order‐by‐order many‐body perturbation theory (MBPT) expansion for the expectation values and density matrices. We also show how the commutator expansion derived by us can be used as a basis for size‐extensive infinite‐order summation techniques. An operator technique of eliminating the nonlocal, “off‐energy shell” denominators from MBPT expressions is proposed and applied to obtain compact commutator formulas for the expectation values of one‐ and two‐electron operators through the fourth and third order, respectively, and for the correlation energy through the fifth order of perturbation theory. © 1993 John Wiley & Sons, Inc.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.