Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ® package

We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender–Wu,and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use Mathematica ® package we call BenderWu. Our package enables quick home-computer computation of high orders of perturbation theory (about 100 orders in 10–30 s, and 250 orders in 1–2 h) and enables practical study of a large class of problems in Quantum Mechanics. We have two hopes concerning the BenderWu package. One is that due to resurgence, large amount of non-perturbative information, such as non-perturbative energies and wave-functions (e.g. WKB wave functions), can in principle be extracted from the perturbative data. We also hope that the package may be used as a teaching tool, providing an effective bridge between perturbation theory and non-perturbative physics in textbooks. Finally, we show that for the multi-variable case, the recursion relation acquires a geometric character, and has a structure which allows parallelization to computer clusters. Program Title: BenderWu Program Files doi: http://dx.doi.org/10.17632/vpg2zsbryc.1 Licensing provisions: CC By 4.0 Programming language: Wolfram Mathematica Nature of problem: In 1D quantum mechanics, a perturbative expansions are known to generically be divergent. An analysis of such problems was so far limited to a case-by-case basis. The Mathematica package presented here allows a quick computation and analysis of all such 1D quantum mechanical problems. Solution method: The program uses a general recursive relation, inspired by the works of Bender and Wu [1], which allows quick computation of the perturbative data. References Carl M. Bender, Tai Tsun Wu, Phys. Rev. 184 (1969) 1231–1260 * Items marked with an asterisk are only required for new versions of programs previously published in the CPC Program Library.