Abstract
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.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This work is of theoretical nature and therefore no data exists to store. The codes that led to the figures are available from the authors.]
References
H. Jeffreys, Proc. Lond. Math. Soc. 2(1), 428 (1923)
D. Gough, Astronomische Nachrichten 328(3–4), 273 (2007)
R. Blumel, Advanced Quantum Mechanics: The Classical-Quantum Connection (Jones & Bartlett Publishers, Burlington, 2011)
M. Vandyck, Eur. J. Phys. 12(3), 112 (1991)
S.K. Adhikari, M.S. Hussein, Am. J. Phys. 76(12), 1108 (2008)
M.S. Child, Semiclassical Mechanics with Molecular Applications (Oxford University Press, New York, 2014)
M.V. Berry, K. Mount, Rep. Prog. Phys. 35(1), 315 (1972)
A. Amthong, Eur. J. Phys. 35(6), 065009 (2014)
M. Arzamasovs, B. Liu, Eur. J. Phys. 38(6), 065405 (2017)
S.C. Miller Jr., R. Good Jr., Phys. Rev. 91(1), 174 (1953)
S. Miller Jr., Phys. Rev. 94(5), 1345 (1954)
R. Good Jr., Phys. Rev. 90(1), 131 (1953)
M. Berry, A.O. de Almeida, J. Phys. A 6(10), 1451 (1973)
G. Wang, A. Chernikov, M.M. Glazov, T.F. Heinz, X. Marie, T. Amand, B. Urbaszek, Rev. Mod. Phys. 90, 021001 (2018)
A. Chaves, R. Ribeiro, T. Frederico, N. Peres, 2D Mater. 4(2), 025086 (2017)
K. Zhang, Y. Feng, F. Wang, Z. Yang, J. Wang, J. Mater. Chem. C 5(46), 11992 (2017)
J.D. Caldwell, I. Aharonovich, G. Cassabois, J.H. Edgar, B. Gil, D. Basov, Nat. Rev. Mater. 4(8), 552 (2019)
J. Henriques, G. Ventura, C. Fernandes, N. Peres, J. Phys. Condens. Matter 32(2), 025304 (2020)
S. Rytova, Mosc. Univ. Phys. Bull. 22, 30 (1967)
L. Keldysh, Sov. J. Exp. Theor. Phys. Lett. 29, 658 (1979)
T. Galvani, F. Paleari, H.P.C. Miranda, A. Molina-Sánchez, L. Wirtz, S. Latil, H. Amara, F.M.C. Ducastelle, Phys. Rev. B 94, 125303 (2016)
A.K. Ghatak, R. Gallawa, I. Goyal, NIST 92, 20427 (1991)
I. Goyal, R. Gallawa, A. Ghatak, J. Electromagn. Waves Appl. 5(6), 623 (1991)
J.P. Dahl, W.P. Schleich, J. Phys. Chem. A 108(41), 8713 (2004)
A. Sinha, R. Roychoudhury, Y. Varshni, Phys. B Cond. Matt. 325, 214 (2003)
M.V. Berry, K. Burke, Eur. J. Phys. 40(6), 065403 (2019)
X. Yang, S. Guo, F. Chan, K. Wong, W. Ching, Phys. Rev. A 43(3), 1186 (1991)
S. Ullah, F. Sato, M.G. Menezes, R.B. Capaz, Phys. Rev. B 100, 085427 (2019)
J.C.G. Henriques, G. Catarina, A.T. Costa, J. Fernández-Rossier, N.M.R. Peres, Phys. Rev. B 101, 045408 (2020)
M. Selig, G. Berghäuser, A. Raja, P. Nagler, C. Schüller, T.F. Heinz, T. Korn, A. Chernikov, E. Malic, A. Knorr, Nat. Commun. 7, 13279 (2016)
Acknowledgements
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.
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Henriques, J.C.G., Peres, N.M.R. Exciton energies and wave functions in hexagonal boron nitride using Miller and Good’s uniform approach. Eur. Phys. J. B 94, 7 (2021). https://doi.org/10.1140/epjb/s10051-020-00014-6
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DOI: https://doi.org/10.1140/epjb/s10051-020-00014-6