Refined Madelung Equations

Abstract

The Madelung equations are two equations that are equivalent to the one-body time-dependent Schrödinger equation. In this paper, the Madelung equation, whose gradient is an Euler equation, is refined by introducing interpretations of functions that are shown to depend only on the realpart of the complex-valued wavefunction. These interpretations are extensions of functions from the recently derived generalized Bernoulli equation, applicable to real-valued quantum-mechanical stationary states. In particular, the velocity and pressure definitions are extended so that they depend on the real-part of a time-dependent complex-valued wavefunction. The Bohn quantum

A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions

“…. Also, the integrand of the quantum-mechanical expectation value of the kinetic energy, given by (12), is interpreted as a sum of two terms: The kinetic energy per volume ρ m u 2 /2 and the pressure p. Elsewhere [43], an extension of these interpretations are used to refine, and further develop, the Madelung equations, where the quantum potential is replaced by an additional kinetic energy term and a term involving the pressure. The main equation of the formalism reduces to the Bernoullian equation (3) for stationary states with real valued wavefunctions.…”

“…Furthermore, the integrand of the quantum-mechanical expectation value of the kinetic energy is given as a sum of two terms: One term is interpreted as the kinetic energy per volume and the other one is the pressure, and the kinetic energy interpretation agrees with an interpretation given by Salesi [41]. Elsewhere [43] an extension of these interpretations are used to extend the applicability range of the Madelung equations. Also, in this paper (in section 3), it is demonstrated that for many quantum mechanical stationary states, including the hydrogen atom s states, local mass conservation is incompatible with steady-flow.…”

“…Elsewhere [43] an extension of the interpretations given in this work are used to refine, and further develop, the Madelung equations. In particular, equation (3) is modified by the addition of a kinetic energy term mv 2 /2 and the replacement of Ē by the time t derivative term − ∂S/∂t, where the Madelung equations use the wavefunction ansatz f r = e iS and the velocity definition v = ∇S/m, and where S = S(r, t).…”