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Journal of Physics A Mathematical and General

Variational and norm criteria in high-order perturbative-variational wavefunctions

F. Arias de Saavedra, Enrique Buendía

Published Jun 7, 1989• 8 citations
Publisher

Abstract

The behaviour of high-order perturbative-variational wavefunctions, with one or more free parameters, is studied. A variational criterion, and a criterion based upon variation of the norm of the perturbative approximation function with order, are used for fixing free parameters. For the quartic anharmonic oscillator, both criteria seem to be equivalent. The obtained energies, not only for the ground state but also for excited states, are in excellent agreement with previous results, although when the authors go to higher excited states they lose precision because of the worsening of the perturbative-variational wavefunctions.

DOI
10.1088/0305-4470/22/11/030

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The behaviour of high-order perturbative-variational wavefunctions, with one or more free parameters, is studied. A variational criterion, and a criterion based upon variation of the norm of the perturbative approximation function with order, are used for fixing free parameters. For the quartic anharmonic oscillator, both criteria seem to be equivalent. The obtained energies, not only for the ground state but also for excited states, are in excellent agreement with previous results, although when the authors go to higher excited states they lose precision because of the worsening of the perturbative-variational wavefunctions.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
In this work, an unusual approach is taken to calculate the ground state and first excited state energies of the anharmonic oscillator potential. By using the variational method, analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators are obtained. The corrections due to the other terms are then calculated perturbatively. It is shown that each approach, which starts with a different choice of zeroth‐order term, provides results that agree with those computed from numerical integration and series solution. Joining the above schemes, yields a combined variational and perturbative treatment of anharmonic oscillator problems.
The direct point-by-point application of minimal sensitivity optimisation to the first-order perturbative approximation to the ground-state wavefunction of the quartic oscillator, carried out previously, is extended to low-lying excited states. An analytic formula for the derivative with respect to the redundant parameter is developed and used to obtain the minimally sensitive value for this parameter as a function of configuration space. Generally, more than one such minimally sensitive value is found at each configuration space point-these values are categorised into disjoint continuous curves. For all the cases studied, it has proved possible to piece together sections of such curves in an unambiguous manner so as to obtain optimised wavefunction approximations which are both renormalisable and continuously differentiable. Comparison of these optimised wavefunction approximations with the exact results revealed satisfactory agreement, particularly in the large mod x mod asymptotic region where orthodox perturbation methods fail completely due to the strong coupling.
It is observed that variationally calculated wavefunctions do not satisfy orthogonality relation when tested for the potentials like anharmonic oscillator and deviated hydrogen atom type. A new approach is suggested using quasi-states only to correlate between variationally calculated wavefunctions and orthogonality relation using perturbation theory considering the anharmonic oscillator example. Analytically expression for energy levels of the anharmonic oscillator are calculated up to the second order. A test for convergency of the perturbation theory is also discussed.
We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.
Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models of their important counterparts in a higher number of dimensions, and as algebraically simple yet highly nontrivial problems on which various approximation or truncation schemes could be tested to very high orders. Such techniques have included perturbation theory and its many possible resummation schemes, and the coupled-cluster method. In the present paper the spotlight is focused on variational methods, and we concentrate on their application to the particularly demanding double-well quartic anharmonic oscillator. We examine in detail both single-state and multistate variational calculations based on either orthogonal (uncorrelated) or nonorthogonal (correlated) basis functions. Particular attention is paid to the question of the sensitivity and accuracy of the results with respect to any nonlinear parameters that characterize the basis states. Practical schemes are suggested for their choice and are shown to be capable of very high accuracy. Use of this simple model also allows detailed comparisons to be made with other techniques of a perturbative or perturbative-variational hybrid character.