A new hydrodynamic formulation of complex-valued quantum mechanics

Ciann-Dong Yang

Chaos, Solitons & Fractals, 2009, vol. 42, issue 1, 453-468

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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:42:y:2009:i:1:p:453-468

DOI: 10.1016/j.chaos.2009.01.010

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