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 wavefunctionW(z,t), z ¼ x þ iy 2 C, we first define a complex potential function X (z,t) = ⁄/(im) lnW(z,t) = /(x,y,t) + iw(x,y,t) with x; y 2 R and then prove that the streamline lines w(x,y,t) = cw and the potential lines / (x,y,y) = c/ in the potential flow defined by X are equivalent to the constant-probability lines jWj = c1 and the constant-phase lines \W= c2 in the probability flow defined by W. 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.