Abstract
We analyze, in general terms, the evolution of energy levels in quantum mechanics, as a function of a coupling parameter, and demonstrate the possibility of level crossings in systems described by irreducible matrices. In long-range interactions, the coupling parameter is the interatomic distance. We demonstrate the utility of adjacency matrices and adjacency graphs in the analysis of “hidden” symmetries of a problem; these allow us to break reducible matrices into irreducible subcomponents. A possible breakdown of the no-crossing theorem for higher-dimensional irreducible matrices is indicated, and an application to the 2S–2S interaction in hydrogen is briefly described. The analysis of interatomic interactions in this system is important for further progress on optical measurements of the 2S hyperfine splitting.
This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.
References
C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, vol. 1, 1st edn. (Wiley, New York, 1978)
C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, vol. 2, 1st edn. (Wiley, New York, 1978)
U. D. Jentschura, V. Debierre, C. M. Adhikari, A. Matveev, and N. Kolachevsky, Long-range interactions of excited hydrogen atoms. II. Hyperfine-resolved (\( 2S;2S \))-system, submitted to Physical Review A (2016)
S. Wolfram, The Mathematica Book, 4th edn. (Cambridge University Press, Cambridge, 1999)
See the URL http://en.m.wikipedia.org/wiki/ Avoided_crossing
L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Volume 3 of the Course on Theoretical Physics (Pergamon Press, Oxford, 1958)
S. Jonsell, A. Saenz, P. Froelich, R.C. Forrey, R. Côté, A. Dalgarno, Long-range interactions between two 2s excited hydrogen atoms. Phys. Rev. A 65, 042501 (2002)
S.I. Simonsen, L. Kocbach, J.P. Hansen, Longrange interactions and state characteristics of interacting Rydberg atoms. J. Phys. B 44, 165001 (2011)
C. Itzykson, J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)
S.R. Lundeen, F.M. Pipkin, Measurement of the Lamb shift in hydrogen, n=2. Rev. Lett. 46, 232–235 (1981)
M. Fischer, N. Kolachevsky, S.G. Karshenboim, T. W. Hänsch, Optical measurement of the \( 2S \) hyperne interval in atomic hydrogen. Can. J. Phys. 80, 1225–1231 (2002)
N. Kolachevsky, M. Fischer, S.G. Karshenboim, T.W. Hänsch, High-precision optical measurement of the \( 2S \) hyperfine interval in atomic hydrogen. Phys. Rev. Lett. 92, 033003 (2004)
N. Kolachevsky, A. Matveev, J. Alnis, C.G. Parthey, S.G. Karshenboim, T.W. Hänsch, Measurement of the \( 2S \) hyperfine interval in atomic hydrogen. Phys. Rev. Lett. 102, 213002 (2009)
Acknowledgements
The authors acknowledge support from the National Science Foundation (Grant PHY–1403973).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 The Author(s)
About this chapter
Cite this chapter
Adhikari, C.M., Debierre, V., Jentschura, U.D. (2018). Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-degenerate States: Applied Graph Theory. In: Meschede, D., Udem, T., Esslinger, T. (eds) Exploring the World with the Laser. Springer, Cham. https://doi.org/10.1007/978-3-319-64346-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-64346-5_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64345-8
Online ISBN: 978-3-319-64346-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)