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A new hydrodynamic formulation of complex-valued quantum mechanics

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  • Yang, Ciann-Dong

Abstract

In this paper, a new hydrodynamic formulation of complex-valued quantum mechanics is derived to reveal a novel analogy between the probability flow and the potential flow on the complex plane. For a given complex-valued wavefunction Ψ(z,t), z=x+iy∈C, we first define a complex potential function Ω (z,t)=ℏ/(im)lnΨ(z,t)=ϕ(x,y,t)+iψ(x,y,t) with x,y∈R and then prove that the streamline lines ψ(x,y,t)=cψ and the potential lines ϕ(x,y,y)=cϕ in the potential flow defined by Ω are equivalent to the constant-probability lines ∣Ψ∣=c1 and the constant-phase lines ∠Ψ=c2 in the probability flow defined by Ψ. The discovered analogy is very useful in visualizing the unobservable probability flow on the complex x+iy plane by analogy with the 2D potential flow on the real x−y plane, which can be visualized by using dye streaks in a fluid laboratory.

Suggested Citation

  • Yang, Ciann-Dong, 2009. "A new hydrodynamic formulation of complex-valued quantum mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 453-468.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:1:p:453-468
    DOI: 10.1016/j.chaos.2009.01.010
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    References listed on IDEAS

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    1. Yang, Ciann-Dong & Wei, Chia-Hung, 2007. "Parameterization of all path integral trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 118-134.
    2. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    3. Yang, Ciann-Dong & Wei, Chia-Hung, 2008. "Strong chaos in one-dimensional quantum system," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 988-1001.
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    Cited by:

    1. 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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