Quasilinearization Method and WKB

TL;DR

The paper compares the quasilinearization method (QLM) with WKB for solving the Schrödinger equation, showing QLM's higher accuracy, absence of singularities, and rapid convergence, making it a potentially superior approach.

Contribution

It introduces the application of QLM to quantum problems, demonstrating its advantages over WKB in accuracy, convergence speed, and handling of turning points.

Findings

First QLM iteration yields ~a few percent accuracy.

Higher QLM iterations rapidly improve accuracy, reaching 20 significant figures.

QLM avoids unphysical singularities present in WKB.

Abstract

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers of h, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schroedinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.