IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v42y2009i2p711-723.html
   My bibliography  Save this article

Stability and quantization of complex-valued nonlinear quantum systems

Author

Listed:
  • Yang, Ciann-Dong

Abstract

In this paper, we show that quantum mechanical systems can be fully treated as complex-extended nonlinear systems such that stability and quantization of the former can be completely analyzed by the existing tools developed for the latter. The concepts of equilibrium points, index theory and Lyapunov stability theory are all extended to a complex domain and then used to determine the stability of quantum mechanical systems. Modeling quantum mechanical systems by complex-valued nonlinear equations leads naturally to the phenomenon of quantization. Based on the residue theorem, we show that the quantization of a physical quantity f(x,p) is a consequence of the trajectory independence of its time-averaged mean value 〈f(x,p)〉x(t). Three types of trajectory independence are observed in quantum systems. Local and global trajectory independences correspond to the quantizations of 〈f(x,p)〉x(t) within a given state ψ, while universal trajectory independence implies that 〈f(x,p)〉x(t) is further independent of the quantum state ψ. By using the property of universal trajectory independence, we give a formal proof of the Bohr–Sommerfeld quantization postulate ∫pdx=nh and the Planck–Einstein quantization postulate E=nhν, n=0,1,….

Suggested Citation

  • Yang, Ciann-Dong, 2009. "Stability and quantization of complex-valued nonlinear quantum systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 711-723.
  • Handle: RePEc:eee:chsofr:v:42:y:2009:i:2:p:711-723
    DOI: 10.1016/j.chaos.2009.01.044
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077909000393
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2009.01.044?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Yang, Ciann-Dong, 2008. "Complex dynamics in diatomic molecules. Part I: Fine structure of internuclear potential," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 962-976.
    2. Yang, Ciann-Dong & Wei, Chia-Hung, 2007. "Parameterization of all path integral trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 118-134.
    3. Yang, Ciann-Dong, 2007. "Quantum motion in complex space," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1073-1092.
    4. Yang, Ciann-Dong, 2007. "Complex tunneling dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 312-345.
    5. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    6. Yang, Ciann-Dong, 2008. "Spin: Nonlinear zero dynamics of orbital motion," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1158-1171.
    7. Yang, Ciann-Dong, 2006. "On modeling and visualizing single-electron spin motion," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 41-50.
    8. Yang, Ciann-Dong & Wei, Chia-Hung, 2008. "Strong chaos in one-dimensional quantum system," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 988-1001.
    9. Yang, Ciann-Dong & Weng, Hung- Jen, 2008. "Complex dynamics in diatomic molecules. Part II: Quantum trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 16-35.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zeng, Xu & Li, Chuandong & Huang, Tingwen & He, Xing, 2015. "Stability analysis of complex-valued impulsive systems with time delay," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 75-82.
    2. Yang, Ciann-Dong & Weng, Hung-Jen, 2012. "Nonlinear quantum dynamics in diatomic molecules: Vibration, rotation and spin," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 402-415.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yang, Ciann-Dong, 2008. "On the existence of complex spacetime in relativistic quantum mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 316-331.
    2. Yang, Ciann-Dong, 2009. "A new hydrodynamic formulation of complex-valued quantum mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 453-468.
    3. Yang, Ciann-Dong & Weng, Hung-Jen, 2012. "Nonlinear quantum dynamics in diatomic molecules: Vibration, rotation and spin," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 402-415.
    4. Yang, Ciann-Dong, 2009. "Complex spin and anti-spin dynamics: A generalization of de Broglie–Bohm theory to complex space," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 317-333.
    5. Yang, Ciann-Dong, 2007. "The origin and proof of quantization axiom p→pˆ=-iℏ∇ in complex spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 274-283.
    6. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.
    7. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    8. Yang, Ciann-Dong & Wei, Chia-Hung, 2007. "Parameterization of all path integral trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 118-134.
    9. El Naschie, Mohamed Saladin, 2006. "The idealized quantum two-slit gedanken experiment revisited—Criticism and reinterpretation," Chaos, Solitons & Fractals, Elsevier, vol. 27(4), pages 843-849.
    10. Iovane, Gerardo & Giordano, Paola, 2007. "Wavelets and multiresolution analysis: Nature of ε(∞) Cantorian space–time," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 896-910.
    11. Alimohammady, M. & Roohi, M., 2007. "Linear minimal space," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1348-1354.
    12. Onbaşlı, Ülker & Özdemir, Zeynep Güven & Aslan, Özden, 2009. "Symmetry breakings and topological solitons in mercury based d-wave superconductors," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1980-1989.
    13. Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.
    14. Falcón, Sergio & Plaza, Ángel, 2008. "On the 3-dimensional k-Fibonacci spirals," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 993-1003.
    15. Alimohammady, M. & Roohi, M., 2009. "Extreme points in minimal spaces," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1480-1485.
    16. Falcon, Sergio & Plaza, Ángel, 2009. "k-Fibonacci sequences modulo m," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 497-504.
    17. Akbulak, Mehmet & Bozkurt, Durmuş, 2009. "On the order-m generalized Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1347-1355.
    18. El Naschie, M.S., 2006. "Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 39-42.
    19. Abu-Donia, H.M., 2007. "Common fixed point theorems for fuzzy mappings in metric space under ϕ-contraction condition," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 538-543.
    20. Mukhamedov, A.M., 2009. "Meso-structures of dynamical chaos and E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1930-1938.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:42:y:2009:i:2:p:711-723. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.