Transition Decomposition of Quantum Mechanical Evolution

Y. Strauss; J. Silman; S. Machnes; L. P. Horwitz

doi:10.1007/s10773-011-0689-y

Transition Decomposition of Quantum Mechanical Evolution

Y. Strauss
, J. Silman
, S. Machnes
, L. P. Horwitz

Department of Physics - at Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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.

Original language	English
Pages (from-to)	2179-2190
Number of pages	12
Journal	International Journal of Theoretical Physics
Volume	50
Issue number	7
DOIs	https://doi.org/10.1007/s10773-011-0689-y
State	Published - Jul 2011

Bibliographical note

Funding Information:
Acknowledgements Y. Strauss acknowledges support from the Israeli Science Foundation (Grants No. 1169/06 and 1105/10). J. Silman and S. Machnes acknowledge support from the Israeli Science Foundation (Grant No. 784/06). J. Silman also acknowledges the support of the Inter-University Attraction Poles Programme (Belgian Science Policy) under Project IAP-P6/10 (Photonics@be) and of the FNRS.

Funding

Acknowledgements Y. Strauss acknowledges support from the Israeli Science Foundation (Grants No. 1169/06 and 1105/10). J. Silman and S. Machnes acknowledge support from the Israeli Science Foundation (Grant No. 784/06). J. Silman also acknowledges the support of the Inter-University Attraction Poles Programme (Belgian Science Policy) under Project IAP-P6/10 (Photonics@be) and of the FNRS.

Funders	Funder number
Israeli Science Foundation	1169/06, 1105/10, 784/06, IAP-P6/10
Fonds De La Recherche Scientifique - FNRS

Keywords

Lyapunov operator
Resonance
Semigroup decomposition
Transition decomposition

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10.1007/s10773-011-0689-y

Cite this

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title = "Transition Decomposition of Quantum Mechanical Evolution",

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.",

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note = "Funding Information: Acknowledgements Y. Strauss acknowledges support from the Israeli Science Foundation (Grants No. 1169/06 and 1105/10). J. Silman and S. Machnes acknowledge support from the Israeli Science Foundation (Grant No. 784/06). J. Silman also acknowledges the support of the Inter-University Attraction Poles Programme (Belgian Science Policy) under Project IAP-P6/10 (Photonics@be) and of the FNRS.",

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N2 - 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.

AB - 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.

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KW - Semigroup decomposition

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Transition Decomposition of Quantum Mechanical Evolution

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