Exciton energies and wave functions in hexagonal Boron Nitride using Miller and Good’s uniform approach

In this paper we revisit the work of Miller and Good, which describes an uniform JWKB type of approximation to the solution of quantum problems. This paper, very well known in atomic physics in the 1970's 1980's of the last century, did not attract the same attention from the condensed matter community. Contrary to the usual JWKB approach, Miller and Good's method yields wave functions that do not diverge at the classical turning points. We apply the method in the context of two-dimensional excitons, an important condensed matter system. In particular, we apply our results to excitons in hexagonal boron nitride, solving the corresponding Wannier equation. We compare the semiclassical results with others from the literature and find good agreement. The authors acknowledge Mike Belsley for carefully reading the manuscript and for comments that allowed to improve the overall presentation. N.M.R.P. acknowledges support from the European Commission through the project "Graphene-Driven Revolutions in ICT and Beyond" (Ref. no. 881603 - core 3), and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2019. In addition, N.M.R.P. acknowledges COMPETE2020, PORTUGAL2020, FEDER and the Portuguese Foundation for Science and Technology (FCT) through projects POCI-01-0145-FEDER-028114, POCI-01-0145-FEDER- 029265, PTDC/NAN-OPT/29265/2017, and POCI-01-0145-FEDER-02888.