Journal of Mathematical Chemistry

, Volume 50, Issue 5, pp 1079–1090 | Cite as

Rényi entropy of the infinite well potential in momentum space and Dirichlet-like trigonometric functionals

  • A. I. Aptekarev
  • J. S. Dehesa
  • P. Sánchez-Moreno
  • D. N. Tulyakov
Original Paper


The momentum entropic moments and Rényi entropies of a one-dimensional particle in an infinite well potential are found by means of explicit calculations of some Dirichlet-like trigonometric integrals. The associated spreading lengths and quantum uncertainty-like sums are also provided.


Information-theoretic measures Quantum infinite well Rényi entropy Rényi spreading length 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (2004)Google Scholar
  2. 2.
    J.B.M. Uffink, Measures of Uncertainty and the Uncertainty Principle. PhD Thesis, University of Utrecht, 1990. See also references hereinGoogle Scholar
  3. 3.
    Frieden B.R.: Science from Fisher Information. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  4. 4.
    Hall M.J.W.: Universal geometric approach to uncertainty, entropy and information. Phys. Rev. A 59, 2602–2615 (1999)CrossRefGoogle Scholar
  5. 5.
    Rényi A.: Probability Theory. North Holland, Amsterdam (1970)Google Scholar
  6. 6.
    C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  7. 7.
    Tsallis C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)CrossRefGoogle Scholar
  8. 8.
    Parr R.G., Yang W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York (1989)Google Scholar
  9. 9.
    Dehesa J.S., López-Rosa S., Martínez-Finkelshtein A., Yáñez R.J.: Information theory of d-dimensional hydrogenic systems: application to circular and Rydberg states. Int. J. Quantum Chem. 110, 1529–1548 (2010)Google Scholar
  10. 10.
    Dehesa J.S., van Assche W., Yáñez R.J.: Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom. Phys. Rev. A 50, 3065–3079 (1994)CrossRefGoogle Scholar
  11. 11.
    Bialynicki-Birula I.: Formulations of uncertainty relations in terms of Rényi entropies. Phys. Rev. A 74, 052101 (2006)CrossRefGoogle Scholar
  12. 12.
    Dehesa J.S., Yáñez R.J., Aptekarev A.I., Buyarov V.S.: Strong asymptotics of Laguerre polynomials and information entropies of two-dimensional harmonic oscillator and one-dimensional Coulomb potentials. J. Math. Phys. 39, 3050–3060 (1998)CrossRefGoogle Scholar
  13. 13.
    López-Rosa S., Montero J., Sánchez-Moreno P., Venegas J., Dehesa J.S.: Position and momentum information-theoretic measures of a d-dimensional particle-in-a-box. J. Math. Chem. 49, 971–994 (2011)CrossRefGoogle Scholar
  14. 14.
    Bohr A., Mottelson B.R.: Nuclear Structure. World Scientific, Singapore (1998)Google Scholar
  15. 15.
    Pederson T.G., Johansen P.M., Pederson H.C.: Particle-in-a-box model of one-dimensional excitons in conjugated polymers. Phys. Rev. B 61, 10504–10510 (2000)CrossRefGoogle Scholar
  16. 16.
    Rubio A., Sánchez-Portal D., Artacho E., Ordejón P., Soler J.M.: Electronic states in a finite carbon nanotube: a one-dimensional quantum box. Phys. Rev. Lett. 82, 3520–3523 (1999)CrossRefGoogle Scholar
  17. 17.
    Harrison P.: Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures. Wiley, New York (2005)CrossRefGoogle Scholar
  18. 18.
    Hu B., Li B., Liu J., Gu Y.: Quantum chaos of a kicked particle in an infinite well potential. Phys. Rev. Lett. 82, 4224 (1999)CrossRefGoogle Scholar
  19. 19.
    Majernik V., Charvot R., Majernikova E.: The momentum entropy of the infinite potential well. J. Phys. A. Math. Gen. 32, 2207 (1999)CrossRefGoogle Scholar
  20. 20.
    Majernik V., Richterek L.: Entropic uncertainty relations for the infinite well. J. Phys. A. Math. Gen. 30, L49–L54 (1997)CrossRefGoogle Scholar
  21. 21.
    Sánchez-Ruiz J.: Asymptotic formula for the quantum entropy of position in energy eigenstates. Phys. Lett. A 226, 7 (1997)CrossRefGoogle Scholar
  22. 22.
    Sánchez-Ruiz J.: Asymptotic formulae for the quantum Renyi entropies of position: application to the infinite well. J. Phys. A. Math. Gen. 32, 3419–3432 (1999)CrossRefGoogle Scholar
  23. 23.
    Dym H., McKean H.P.: Fourier Series and Integrals. Academic Press, New York (1972)Google Scholar
  24. 24.
    Catalan R.G., Garay J., López-Ruiz R.: Features of the extension of a statistical measure of complexity to continuous systems. Phys. Rev. E 66, 011102 (2002)CrossRefGoogle Scholar
  25. 25.
    Romera E., Nagy A.: Fisher-R ényi entropy product and information plane. Phys. Lett. A 372, 6823–6825 (2008)CrossRefGoogle Scholar
  26. 26.
    López-Ruiz R., Sañudo J.: Complexity invariance by replication in the quantum square well. Open Syst. Inf. Dyn. 16, 423–427 (2009)CrossRefGoogle Scholar
  27. 27.
    Nagy A., Sen K.D., Montgomery H.E. Jr.: LMC complexity for the ground state of different quantum systems. Phys. Lett. A 373, 2552–2555 (2009)CrossRefGoogle Scholar
  28. 28.
    Sen K.D.: Statistical Measures: Applications to Electronic Structure. Springer, Berlin (2011)Google Scholar
  29. 29.
    Abramowitz, M., Stegun, I.A. (eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standars, U.S. Goverment Printing Office, Washington, D.C. (1964)Google Scholar
  30. 30.
    Zozor S., Portesi M., Vignat C.: Some extensions of the uncertainty principle. Phys. A 387, 19–20 (2008)Google Scholar
  31. 31.
    Maassen H., Uffink J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)CrossRefGoogle Scholar
  32. 32.
    Rajagopal A.K.: The Sobolev inequality and the Tsallis entropic uncertainty relation. Phys. Lett. A 205, 32–36 (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • A. I. Aptekarev
    • 1
  • J. S. Dehesa
    • 2
    • 3
  • P. Sánchez-Moreno
    • 2
    • 4
  • D. N. Tulyakov
    • 1
  1. 1.Keldysh Institute for Applied Mathematics, Russian Academy of SciencesMoscow State UniversityMoscowRussia
  2. 2.Institute “Carlos I” for Computational and Theoretical PhysicsUniversity of GranadaGranadaSpain
  3. 3.Department of Atomic, Molecular and Nuclear PhysicsUniversity of GranadaGranadaSpain
  4. 4.Department of Applied MathematicsUniversity of GranadaGranadaSpain

Personalised recommendations