Convergence of transition amplitudes obtained with the Schwinger variational principle

TL;DR

This paper investigates the convergence behavior of transition amplitudes calculated via the Schwinger variational principle in a solvable quantum problem, highlighting its advantages and limitations compared to perturbative methods.

Contribution

It provides a detailed comparison of the Schwinger variational principle with perturbation theory, demonstrating its improved performance and discussing convergence issues.

Findings

Schwinger variational principle outperforms perturbative series in approximating transition amplitudes.

Transition amplitudes do not converge for large perturbations at fixed order.

Using better trial wave functions and higher order improves the method's accuracy.

Abstract

An exactly solvable time-dependent quantum mechanical problem is employed to study the convergence properties of transition amplitudes calculated by using the Schwinger variational principle. A detailed comparison between the amplitudes approximated by the perturbative series and by their associated Schwinger variational principles is performed. The much better performance obtained by the variational principle is documented through different case studies. For a given order of the Schwinger principle, it is observed that the transition amplitudes do not converge to the exact one for large perturbations. The latter is true even though large combinations of unperturbed states with constant coefficients are taken as trial wave functions. As a matter of fact, it is shown that the improvement of the method comes from using better trial wave functions and increasing the order of the Schwinger principle employed.