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Stability and quantization of complex-valued nonlinear quantum systems

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  • 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
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    References listed on IDEAS

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    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, 2007. "Complex tunneling dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 312-345.
    3. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    4. Yang, Ciann-Dong & Wei, Chia-Hung, 2007. "Parameterization of all path integral trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 118-134.
    5. Yang, Ciann-Dong, 2007. "Quantum motion in complex space," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1073-1092.
    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.
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    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.

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