Foundations of Physics

, Volume 45, Issue 2, pp 177–197 | Cite as

A Discussion on the Properties of Gamow States

Article

Abstract

Gamow states are vector states for the pure decaying part of a quantum resonance. We review and analyze the properties of Gamow vectors in different representations. In particular, we discuss the controversial problem of assigning a mean value of the energy for a Gamow state from several points of view. The question on whether a Gamow state is a pure state or not is also analyzed here, as has relevance on the assignation of a non-zero value for the entropy for a Gamow state.

Keywords

Quantum resonances Gamow states Energy averages on Gamow states 

References

  1. 1.
    Fonda, L., Ghirardi, G.C., Rimini, A.: Decay theory of unstable quantum systems. Rep. Progr. Phys. 41, 588–631 (1978)ADSCrossRefGoogle Scholar
  2. 2.
    Fischer, M.C., Gutierrez-Medina, B., Raizen, M.G.: Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys. Rev. Lett. 87, 040402 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    Rothe, C., Hintschich, S.L., Monkman, A.P.: Violation of the exponential-decay law at long times. Phys. Rev. Lett. 96, 163601 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    Nakanishi, N.: A theory of clothed unstable particles. Progr. Theor. Phys. 19, 607–621 (1958)ADSCrossRefMATHGoogle Scholar
  5. 5.
    Bohm, A.: The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics, vol. 78. Springer, New York (1978)CrossRefGoogle Scholar
  6. 6.
    Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions: Applications of Harmonic Analysis. Academic, New York (1964)Google Scholar
  7. 7.
    Roberts, J.E.: Rigged Hilbert spaces in quantum mechanics. Commun. Math. Phys. 3, 98–119 (1966)ADSCrossRefMATHGoogle Scholar
  8. 8.
    Antoine, J.P.: Dirac formalism and symmetry problems in quantum mechanics: General Dirac formalism. J. Math. Phys. 10, 53–69 (1969)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Melsheimer, O.: Rigged Hilbert spaces as an extended mathematical formalism for quantum mechanics: 1 General theory. J. Math. Phys. 15, 902–916 (1974)Google Scholar
  10. 10.
    Bohm, A.: Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J. Math. Phys. 22, 2813–2823 (1981)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bohm, A., Gadella, M.: Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics, vol. 348. Springer Verlag, Berlin (1989)CrossRefGoogle Scholar
  12. 12.
    Civitarese, O., Gadella, M.: Physical and mathematical aspects of Gamow states. Phys. Rep. 396, 41–113 (2004)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bohm, A.: Quantum Mechanics. Foundations and Applications, 3rd edn. Springer, New York (2001)MATHGoogle Scholar
  14. 14.
    Julve, J., de Urriés, F.J.: Inner products of resonance solutions in 1D quantum barriers. J. Phys. A 43, 175301 (2010)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Julve, J., Turrini, S., de Urriés, F.J.: Inner products of energy eigenstates for a 1-D quantum barrier. Int. J. Theor. Phys. 53, 971–984 (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Castagnino, M., Gadella, M., Gaioli, F., Laura, R.: Gamow vectors and time assymmetry. Int. J. Theor. Phys. 38, 2823–2865 (1999)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Berggren, T.: On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes. Nucl. Phys. A 109, 265–287 (1968)ADSCrossRefGoogle Scholar
  18. 18.
    Berggren, T.: Expectation value of an operator in a resonant state. Phys. Lett. B 373, 1–4 (1996)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    Civitarese, O., Gadella, M., Id Betan, R.: On the mean value of the energy for resonance states. Nucl. Phys. A 660, 255–266 (1999)ADSCrossRefGoogle Scholar
  20. 20.
    Petrosky, T., Prigogine, I., Tasaki, S.: Quantum theory of non-integrable systems. Phys. A 173, 175–242 (1991)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Bollini, C.G., Civitarese, O., de Paoli, A.L., Rocca, M.C.: Gamow states as continuous linear functionals over analytical test functions. J. Math. Phys. 37, 4235–4242 (1996)ADSCrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Exner, P.: Open Quantum Systems and Feynman Integrals. Reidel, Dordrecht (1984)Google Scholar
  23. 23.
    Reed, M., Simon, B.: Analysis of Operators. Academic, New York (1978)MATHGoogle Scholar
  24. 24.
    Antoniou, I.E., Laura, R., Suchanecki, Z., Tasaki, S.: Intrinsic irreversibility of quantum systems with diagonal singularity. Phys. A 241, 737–772 (1997)CrossRefGoogle Scholar
  25. 25.
    van Hove, L.: Energy corrections and persistent perturbation effects in continuous spectra. Physica 21, 901 (1955)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    van Hove, L.: The approach to equilibrium in quantum statistics. Physica 23, 441 (1957)ADSCrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    van Hove, L.: The ergodic behaviour of quantum many-body systems. Physica 25, 268 (1959)ADSCrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Castagnino, M., Gadella, M., Id Betan, R., Laura, R.: The Gamow functional. Phys. Lett. A 282, 245–250 (2001)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Castagnino, M., Gadella, M., Id Betan, R., Laura, R.: Gamow functionals on operator algebras. J. Phys. A 34, 10067–10083 (2001)ADSCrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Friedrichs, K.O.: On the perturbation of continuous spectra. Commun. Appl. Math. 1, 361–406 (1948)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Gadella, M., Pronko, G.P.: The Friedrichs model and its use in resonance phenomena. Fortschr. Phys. 59, 795–859 (2011)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Bohm, A., Gadella, M., Kielanowski, P.: Time asymmetric quantum mechanics. SIGMA 7, 086 (2011)MathSciNetGoogle Scholar
  33. 33.
    Bohm, A.R., Erman, F., Uncu, H.: Resonance phenomena and time asymmetric quantum mechanics. Turk. J. Phys. 35, 209–240 (2011)Google Scholar
  34. 34.
    Antoniou, I., Dmitrieva, L., Kuperin, Y., Melnikov, Y.: Resonances and the extension of dynamics to rigged Hilbert space. Comput. Math. Appl. 34, 399–425 (1997)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Civitarese, O., Gadella, M.: On the concept of entropy for decaying states. Found. Phys. 43, 1275–1294 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Antoniou, I., Pronko, G.P., Karpov, E., Yarevsky, E.: Oscillating decay of an unstable system. Int. J. Theor. Phys. 42, 2403–2421 (2003)CrossRefMATHGoogle Scholar
  37. 37.
    Kukulin, V.I., Krasnapolsky, V.M., Horacek, J.: Theory of Resonances. Principles and Applications. Academia, Praha (1989)CrossRefMATHGoogle Scholar
  38. 38.
    Gyarmati, B., Vertse, T.: On the normalization of Gamow functions. Nucl. Phys. A 160, 523–528 (1971)ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Gadella, M., Laura, R.: Gamow dyads and expectation values. Int. J. Quantum Chem. 81, 307–320 (2001)CrossRefGoogle Scholar
  40. 40.
    Bohm, A.: Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J Math. Phys. 21, 2813–2823 (1981)ADSCrossRefGoogle Scholar
  41. 41.
    Gadella, M.: Gamow vectors: miscellaneous results. J. Phys. Conf. Ser. 128, 012038 (2008)ADSCrossRefGoogle Scholar
  42. 42.
    Bohm, A.R., Loewe, M., Van de Ven, B.: Time asymmetric quantum theory – I. Modifying an axiom of quantum physics. Fortschr. Phys. 51, 551–568 (2003)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departamento de Física Teórica, Atómica y ÓpticaUniversidad de ValladolidValladolidSpain

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