Advertisement

International Journal of Theoretical Physics

, Volume 50, Issue 7, pp 2179–2190 | Cite as

Transition Decomposition of Quantum Mechanical Evolution

  • Y. Strauss
  • J. Silman
  • S. Machnes
  • L. P. Horwitz
Article

Abstract

We show that the existence of the family of self-adjoint Lyapunov operators introduced in Strauss (J. Math. Phys. 51:022104, 2010) allows for the decomposition of the state of a quantum mechanical system into two parts: A backward asymptotic component, which is asymptotic to the state of the system in the limit t→−∞ and vanishes at t→∞, and a forward asymptotic component, which is asymptotic to the state of the system in the limit t→∞ and vanishes at t→−∞. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the evolution of the backward asymptotic component captures the behavior of the resonance. In particular, it provides a spatial probability distribution for the resonance and exhibits its typical decay law.

Keywords

Lyapunov operator Transition decomposition Resonance Semigroup decomposition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982) zbMATHGoogle Scholar
  2. 2.
    Busch, P., Grabowski, M., Lahti, P.J.: Operational Quantum Physics. Springer, Berlin (1995) zbMATHGoogle Scholar
  3. 3.
    Pauli, W.E.: In: Handbuch der Physik. Springer, Berlin (1926) Google Scholar
  4. 4.
    Mackey, G.W.: The Theory of Unitary Group Representations. University of Chicago Press, Chicago (1976) (see, for example) zbMATHGoogle Scholar
  5. 5.
    Strauss, Y.: J. Math. Phys. 51, 022104 (2010) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Strauss, Y., Silman, J., Machnes, S., Horwitz, L.P.: arXiv:0802.2448 [quant-ph]
  7. 7.
    Hegerfeldt, G.C., Muga, J.G.: Phys. Rev. A 82, 012113 (2010) ADSCrossRefGoogle Scholar
  8. 8.
    Strauss, Y.: J. Math. Phys. 46, 102109 (2005) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Strauss, Y., Horwitz, L.P., Volovick, A.: J. Math. Phys. 47, 123505 (2006) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Lax, P.D., Phillips, R.S.: Scattering Theory. Academic Press, San Diego (1967) zbMATHGoogle Scholar
  11. 11.
    Strauss, Y., Horwitz, L.P., Eisenberg, E.: J. Math. Phys. 41, 8050 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Strauss, Y.: Int. J. Theor. Phys. 42, 2285 (2003) zbMATHCrossRefGoogle Scholar
  13. 13.
    Horwitz, L.P., Marchand, J.-P.: Rocky Mt. J. Math. 1, 225 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Misra, B., Sudarshan, E.C.G.: J. Math. Phys. 18, 756 (1977) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Strauss, Y.: J. Math. Phys. 46, 032104 (2005) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    de la Madrid, R., Gadella, M.: Am. J. Phys. 70, 626 (2002) ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Y. Strauss
    • 1
  • J. Silman
    • 2
    • 3
  • S. Machnes
    • 2
    • 4
  • L. P. Horwitz
    • 2
    • 5
    • 6
  1. 1.Einstein Institute of Mathematics, Edmond J. Safra campusThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact SciencesTel-Aviv UniversityTel-AvivIsrael
  3. 3.Laboratoire d’Information QuantiqueUniversité Libre de BruxellesBruxellesBelgium
  4. 4.Institut für Theoretische PhysikUniversität UlmUlmGermany
  5. 5.Department of PhysicsBar-Ilan UniversityRamat-GanIsrael
  6. 6.Department of PhysicsThe Ariel University Center of SamariaArielIsrael

Personalised recommendations