Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-degenerate States: Applied Graph Theory

  • C. M. Adhikari
  • V. Debierre
  • U. D. Jentschura


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.



The authors acknowledge support from the National Science Foundation (Grant PHY–1403973).


  1. 1.
    C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, vol. 1, 1st edn. (Wiley, New York, 1978)zbMATHGoogle Scholar
  2. 2.
    C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, vol. 2, 1st edn. (Wiley, New York, 1978)zbMATHGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    S. Wolfram, The Mathematica Book, 4th edn. (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  5. 5.
  6. 6.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Volume 3 of the Course on Theoretical Physics (Pergamon Press, Oxford, 1958)zbMATHGoogle Scholar
  7. 7.
    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)ADSCrossRefGoogle Scholar
  8. 8.
    S.I. Simonsen, L. Kocbach, J.P. Hansen, Longrange interactions and state characteristics of interacting Rydberg atoms. J. Phys. B 44, 165001 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    C. Itzykson, J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)zbMATHGoogle Scholar
  10. 10.
    S.R. Lundeen, F.M. Pipkin, Measurement of the Lamb shift in hydrogen, n=2. Rev. Lett. 46, 232–235 (1981)ADSCrossRefGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)ADSCrossRefGoogle Scholar
  13. 13.
    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)ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • C. M. Adhikari
    • 1
  • V. Debierre
    • 1
  • U. D. Jentschura
    • 1
  1. 1.Department of PhysicsMissouri University of Science and TechnologyRollaUSA

Personalised recommendations