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Journal of Physics A Mathematical and General

Minimal sensitivity optimisation of perturbatively approximated excited-state wavefunctions of the quartic oscillator

Steven Kenneth Kauffmann, S. M. Perez

Published Dec 21, 1986• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0305-4470/19/18/026

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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.
Stevenson's principle (1981) of minimal sensitivity is applied directly to configuration space wavefunctions calculated in first-order perturbation theory. The redundant parameters, which comprise the essential elements of minimal sensitivity calculations, actually become functions of configuration space, giving the method considerable flexibility and potential accuracy. It is shown that the method is exact in an illustrative class of simple cases and, for the more realistic case of the ground state wavefunction of the quartic oscillator, it is considerably more accurate than the closely related perturbative variational approach, particularly in the asymptotic region.
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.
The authors have justified a point-by-point optimisation procedure for parametrised perturbative approximations to wavefunctions by appealing to a principle of minimal sensitivity. Here it is shown that, if the diagonal matrix elements of the perturbation vanish (which can always be arranged via a simple redefinition of the unperturbed Hamiltonian), then the optimisation procedure is also justified by an underlying stationary principle, with the nth-order perturbative approximations being stationary to that order.
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.
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.
We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.