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Journal of Physics A Mathematical and General

Minimal sensitivity optimisation of perturbative wavefunctions

Steven Kenneth Kauffmann, S. M. Perez

Published Jul 11, 1984• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/17/10/015

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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 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.
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.
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 notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to the quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings.
If the potential is not closely related to a potential that we already know how to solve exactly, the energies and states can still be approximated via the Variational Method. This method is simpler to use than to understand. A simple five-step procedure is given and then illustrated via an example. The principle, in a nutshell, is that if you guess at the ground state and are wrong, the energy you calculate will be higher than the ground state energy, so if you keep guessing and keep getting smaller energies, then you must be getting closer to the ground state wave function. This minimization procedure is called the Variational Method, and it was known prior to quantum mechanics.
Abstract Within the framework of the method of variation of constants, time-dependent perturbation theory is presented in a form that naturally eliminates the undesirable ‘secular ’ terms for a time-independent perturbation, permits an order-by-order calculation of the exact wavefunction for a ‘resonant’ harmonic perturbation and offers a simple route to establish the adiabatic hypothesis in quantum mechanics. The formulation rests on the introduction of a flexibility in the choice of the phase factors associated with the varying amplitudes, together with development of a perturbation procedure that closely resembles the Brillouin-Wigner formalism in the static case, and which exploits the flexibility of the phase factors at each order of the theory.