Madelung Picture Research Articles

Uncertainty Relations in the Madelung Picture Including a Dissipative Environment.

In a recent paper, we have shown how in Madelung's hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment.

Solitary wave and modulational instability in a strongly coupled semiclassical relativistic dusty pair plasma with density gradient

Using a set of fluid equations to describe the inertial dust grain component in a dense collision-less unmagnetised plasma under the influence of weakly relativistic semiclassical electrons and positrons, the propagation of dust-acoustic wave is studied in the strong coupling regime when the dust density is non-uniform. Our main aim is to analyse the role of semiclassical and relativistic environment (frequently encountered in astrophysical context) on various features of strongly coupled dusty plasma. The semiclassical environment of the electrons and positrons is assumed to be described by the Chandrasekhar equation of state. Our second aim is to see the effect of spatial variation of the dust equilibrium density, which is known to occur due to the deformation of the Debye sheath which in turn leads to polarisation force. A new type of nonlinear Schrodinger equation with spatially varying coefficient is deduced and its modulational stability is studied in detail. In the last section, we have taken recourse to Madelung picture to deduce a variable coefficient Korteweg–de Vries equation from this new nonlinear Schrodinger equation.

The Madelung Picture as a Foundation of Geometric Quantum Theory

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recoursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we argue that the Schroedinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by E. Madelung, naturally ground the Schroedinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modelling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.

On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung, have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena and more recently in the modeling of semiconductor devices . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method, which iterates a Schrodinger Madelung picture with a suitable wave function updating mechanism. Therefore several \emph{a priori} bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximating sequences. No uniqueness result is provided.