×

Stability and quantization of complex-valued nonlinear quantum systems. (English) Zbl 1198.81119

Summary: 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 \(\langle f(x,p)\rangle_{x(t)}\). Three types of trajectory independence are observed in quantum systems. Local and global trajectory independences correspond to the quantizations of \(\langle f(x,p)\rangle_{x(t)}\) within a given state \(\psi \), while universal trajectory independence implies that \(\langle f(x,p)\rangle_{x(t)}\) is further independent of the quantum state \(\psi \). By using the property of universal trajectory independence, we give a formal proof of the Bohr-Sommerfeld quantization postulate \(\int pdx=nh\) and the Planck-Einstein quantization postulate \(E=nh\nu\), \(n=0,1,\dots \).
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

81Q65 Alternative quantum mechanics (including hidden variables, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] El Naschie, M.S., A review of applications and results of E-infinity theory, Int J nonlinear sci numer simulat, 8, 1, 11-20, (2007)
[2] El Naschie, M.S., E-infinity theory – some recent results and new interpretations, Chaos solitons fractals, 29, 845-853, (2006)
[3] He, J.H., Nonlinear dynamics and the nobel prize in physics, Int J nonlinear sci numer simulat, 8, 1, 1-4, (2007)
[4] He, J.H.; Liu, Y., A hierarchy of motion in electrospinning process and E-infinity nanotechnology, J polym eng, 28, 101-114, (2008)
[5] Nottale, L., Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems, Chaos solitons fractals, 7, 877-938, (1996) · Zbl 1080.81525
[6] El Naschie, M.S., Deterministic quantum mechanics versus classical mechanical indeterminism, Int J nonlinear sci numer simulat, 8, 1, 5-10, (2007)
[7] El Naschie, M.S., Non-Euclidean spacetime structure and the two-slit experiment, Chaos solitons fractals, 26, 1-6, (2005) · Zbl 1122.81338
[8] El Naschie, M.S., On a fuzzy \(K \ddot{a} \mathit{hler}\)-like manifold which is consistent with the two slit experiment, Int J nonlinear sci numer simulat, 6, 95-98, (2006)
[9] El Naschie, M.S., On the unification of the fundamental forces and complex time in the \(E^{(\infty)}\) space, Chaos solitons fractals, 11, 1149-1162, (2000) · Zbl 1094.81574
[10] Yang, C.D., Quantum dynamics of hydrogen atom in complex space, Ann phys, 319, 399-443, (2005) · Zbl 1105.81032
[11] Yang, C.D., Quantum motion in complex space, Chaos solitons fractals, 33, 1073-1092, (2007) · Zbl 1136.81375
[12] Yang, C.D., Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom, Ann phys, 321, 2876-2926, (2006) · Zbl 1105.81036
[13] Yang, C.D., Complex tunneling dynamics, Chaos solitons fractals, 32, 312-345, (2007) · Zbl 1135.81016
[14] Yang, C.D., Wave-particle duality in complex space, Ann phys, 319, 444-470, (2005) · Zbl 1074.81005
[15] Yang, C.D., On modeling and visualizing single-electron spin motion, Chaos solitons fractals, 30, 41-50, (2006)
[16] Yang, C.D., Spin: the nonlinear zero dynamics of orbital motion, Chaos solitons fractals, 37, 1158-1171, (2008)
[17] Yang, C.D.; Han, S.Y.; Hsiao, F.B., Nonlinear dynamics governing quantum transition behavior, Int J nonlinear sci numer simulat, 8, 3, 397-412, (2007)
[18] Yang, C.D.; Wei, C.H., Parameterization of all path integral trajectories, Chaos solitons fractals, 33, 118-134, (2007) · Zbl 1152.81866
[19] Yang, C.D.; Wei, C.H., Strong chaos in one-dimensional quantum system, Chaos solitons fractals, 37, 988-1001, (2008) · Zbl 1149.81315
[20] Yang, C.D., Trajectory interpretation of the uncertainty principle in 1D systems using complex Bohmian mechanics, Phys lett A, 372, 6240-6253, (2008) · Zbl 1225.81006
[21] Yang, C.D., Complex dynamics in diatomic molecules, part I: fine structure of internuclear potential, Chaos solitons fractals, 37, 962-976, (2008)
[22] Yang, C.D.; Weng, H.J., Complex dynamics in diatomic molecules, part II: quantum trajectory, Chaos solitons fractals, 38, 16-35, (2008)
[23] Sastry, S., Nonlinear systems, (1999), Springer-Verlag New York
[24] Rudin, W., Real and complex analysis, (1986), McGraw-Hill Singapore
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.