The hydrodynamic model, originally proposed by Madelung [Mad27], provides a classical picture for quantum dynamics, namely, that of the flow of an indestructible probability fluid.

Mathematically it consists in representing using polar coordinates as

(6.17) |

Substituting this ansatz into the time-dependent Schrödinger equation with constant mass and separating into real and imaginary parts, gives two equations:

where we introduced the quantum density and the so-called quantum potential :

(6.20) |

In these formulas the prime denotes the one-dimensional space derivative.

The quantum potential arises from the kinetic energy of the Schrödinger equation and creates the ``self-field''. Alternatively can be seen as pressure in a hydrodynamical Navier-Stokes interpretation [Har66], which is related to Nelson's stochastic interpretation of quantum mechanics.

In the classical limit (e.g., WKB approximation [Kol00], [Sch69]) Equation 6.19 becomes the Hamilton-Jacobi equation with principal function .

With the identification of the velocity

(6.21) |

and the flux

(6.22) |

the Equation 6.18 becomes the continuity equation

(6.23) |

and the second equation can be rewritten as an equation of motion (assuming constant mass )

where in the last equation denotes the Lagrangian (also called ``convective'') derivative [Bit00], [SRV89].

In the deBroglie-Bohm interpretation of quantum mechanics a particle has a sharp position at each time moving along a fluid trajectory determined by Equation 6.25 (with ). Evolution in the Bohm picture is ``sharp'', but the initial condition given by the initial location of the particle is a distribution. Tunneling is explained by lowering the barrier through the additional Bohm potential . This is a hidden variables theory which singles out position as a preferred variable. The same can be done for other observables, see [Vin00].

In the stationary case we have

(6.25) |

This gives and

(6.26) | ||

(6.27) |

which expresses conservation of particle number and of energy.

The hydrodynamical formulation can be extended to the mixed state case and was used for quantum mechanical simulation in [BM02], [LW99], [Dal03], [Bit00].

It is pointed out in [Wal94], that the hydrodynamical formulation needs an additional quantization constraint for equivalence with the ``standard'' Schrödinger equation.

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