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Physical Review A

Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations

Timothy C. Wallstrom

Published Mar 1, 1994• 210 citations
Publisher

Abstract

By differentiating the Schr\"odinger equation and separating the real amd imaginary parts, one obtains the Madelung hydrodynamic equations, which have inspired numerous classical interpretations of quantum mechanics. Such interpretations frequently assume that these equations are equivalent to the Schr\"odinger equation, and thus provide an alternative basis for quantum mechanics. This paper proves that this is incorrect: to recover the Schr\"odinger equation, one must add by hand a quantization condition, as in the old quantum theory. The implications for various alternative interpretations of quantum mechanics are discussed.

DOI
10.1103/physreva.49.1613

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By differentiating the Schr"odinger equation and separating the real amd imaginary parts, one obtains the Madelung hydrodynamic equations, which have inspired numerous classical interpretations of quantum mechanics. Such interpretations frequently assume that these equations are equivalent to the Schr"odinger equation, and thus provide an alternative basis for quantum mechanics. This paper proves that this is incorrect: to recover the Schr"odinger equation, one must add by hand a quantization condition, as in the old quantum theory. The implications for various alternative interpretations of quantum mechanics are discussed.
In the paper with the above-noted title, T. C. Wallstrom claims that the description of the particle's motion as a certain "conservative" diffusion is not equivalent to quantum mechanics in spite of the fact that the Madelung "hydrodynamic" equations, which provide the description of such a diffusion, can be converted to the Schroedinger equation. He pointed out that such a stochastic theory can be regarded as equivalent to conventional quantum mechanics only if they can derive from it not just the Madelung equations but also the condition that the circulation of the "probability fluid" is always quantized, which is equivalent the condition for the single-valuedness of the wave function. We, however, show that the single-valuedness of the wave function required in quantum mechanics, is not an auxiliary condition but a property of all local solutions of the Schrodinger equation. Based on the one-to-one correspondence between local solutions of the Schroedinger and the Madelung equations this means that the quantization of the circulation of the "probability fluid" is a property of all solutions of the Madelung equations.
The Madelung equations express the Schrödinger equation as a continuity equation and modified Hamilton–Jacobi equation. These equations are equivalent to the Euler equations for a compressible, potential flow, when classical pressure per unit density is replaced by the quantum potential per unit mass. We extend this hydrodynamic interpretation by quantising a single, spinless, non-relativistic particle constrained to a surface wave with small slope. The wave is distinct from the wave function and, in order to reproduce the Schrödinger equation, it must satisfy the kinematic boundary condition for a free surface advected by twice the Madelung velocity field.
A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.
It is shown that the Bohmian mechanics and the Madelung quantum hydrodynamics are different theories and the latter is a better ontological interpretation of quantum mechanics. A new stochastic interpretation of quantum mechanics is proposed, which is the background of the Madelung quantum hydrodynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum potential. It describes the quantum evolution of the probability density by a convective diffusion with imaginary transport coefficients.
Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined.
Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space.