## 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 + 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} = c_{1} and the constant-phase lines ∠Ψ = c_{2} 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.

Original language | English |
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Pages (from-to) | 453-468 |

Number of pages | 16 |

Journal | Chaos, solitons and fractals |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Oct 15 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)