A new hydrodynamic formulation of complex-valued quantum mechanics

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 + i y ∈ C, we first define a complex potential function Ω (z,t) = ℏ/(im) lnΨ(z,t) = φ{symbol}(x,y,t) + iψ(x,y,t) with x, y ∈ R and then prove that the streamline lines ψ(x,y,t) = c_ψ and the potential lines φ{symbol}(x,y,y) = c_φ{symbol} in the potential flow defined by Ω are equivalent to the constant-probability lines {divides}Ψ{divides} = 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.