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The Aeolian harp sound from the Madelung model of quantum mechanics

Miroslav Pardy

Published Apr 18, 2024• 0 citations
Publisher

Abstract

According to Madelung, Bohm and Vigier, Wilhelm, Rosen and others, the original Schroedinger equation can be transformed into the hydrodynamical system of equations by using the so called Madelung ansatz. We derive in such quantum hydrodynamics, the non-relativistic and relativistic Strouhal number from the so called von Karman's vortex street. The relativistic derivation of this formula follows from the addition formula for velocities. The Strouhal friction tones are generated also during the motion of cosmic rays in relic photon sea, during the motion of bolids in atmosphere, during the Saturn rings motion in the relic black-body sea, during the motion of bodies in superfluid helium and so on.

DOI
10.32370/ia_2024_01_3

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According to Madelung, Bohm and Vigier, Wilhelm, Rosen and others, the original Schroedinger equation can be transformed into the hydrodynamical system of equations by using the so called Madelung ansatz. We derive in such quantum hydrodynamics, the non-relativistic and relativistic Strouhal number from the so called von Karman's vortex street. The relativistic derivation of this formula follows from the addition formula for velocities. The Strouhal friction tones are generated also during the motion of cosmic rays in relic photon sea, during the motion of bolids in atmosphere, during the Saturn rings motion in the relic black-body sea, during the motion of bodies in superfluid helium and so on.
The quantum hydrodynamic-like equations for two real variables (i.e., the phase and the amplitude of the wave function) of the relativistic Klein-Gordon equation are derived in the present paper. The paper also shows that in classical limit the hydrodynamic Klein-Gordon equations lead to the Madelung pseudo-potential [1] as well as to the quantum pseudo potential for a charged particle given by Janossy [2] and by Bialynicki et al [3]. The origin of the non-local interactions of quantum mechanics in the hydrodynamic model are discussed both for free and charged particles.
The Madelung equations are two equations that are equivalent to the one-body time-dependent Schroedinger 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 real-part 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 potential is then interpreted as the sum of two terms, one involving the kinetic energy and the other involving the pressure. Substituting the interpreted quantum-potential into the Madelung equation gives a refined equation containing two kinetic energy terms, a pressure term, and the external potential. It is easily demonstrated that the refined Madelung equation, applied to the hydrogen atom states with a nonzero magnetic quantum number, gives a fluid velocity that contains both a radial component and a free vortex. Hence, the fluid particles have angular momentum and move on streamlines that terminate at infinity. It is also demonstrated that the two velocities from the refined Madelung equation are related: One is the real component and the other is the imaginary component of a complex velocity. Furthermore, an Euler equation for quantum mechanical systems is derived by taking the gradient of the refined Madelung equation.
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.
Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.
An analogy between non-relativistic quantum mechanics in the Madelung formulation and quantum geometrodynamics in the case of the maximally symmetric space is drawn. The equations equivalent to the continuity equation and the hydrodynamic Euler equation describing the evolution of the velocity introduced for the case of hypothetical fluid flow characterizing the cosmological system are obtained. It is shown that the perfect nature of the fluid is broken by the quantum Bohm potential. The quantum potential is calculated in the semi-classical approximation for different forces acting in the system both in standard quantum mechanics and in the minisuperspace model of quantum geometrodynamics. The explicit dependences of the cosmic scale factor on the conformal time, which account for the quantum additive, are found for the empty space with spatial curvature and for a spatially flat universe with dust and radiation.