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CRC Press eBooks

Madelung Fluid Dynamics

Nicolae Mazilu, Maricel Agop, Ioan Mercheş

Published Sep 12, 2019• 0 citations
Publisher

Abstract

After the fundamental works of Erwin Schrödinger, Erwin Madelung came up with a physical interpretation of the wave function, from the point of view of the classical fluids’ dynamics. This chapter is concerned with Madelung’s physical interpretation. The Madelung fluid is one case illustrating how “the forces detemine the velocities”, however at a different level of statistics. The “complex phase” of the wave function is the mark of the solution of nonstationary equation. However, his justification for the existence of the essentially complex wave function can be recognized only as anecdotal. The nonstationary Schrödinger equation admits, besides the classical Galilei group proper, an extra set of symmetries that can be taken in a form involving just one space dimension and time. The Maxwellian form of electrodynamics shows that, as long as the time is classically considered i.e. as just a parameter of continuity this concept of local does not require any further consideration beyond in a given space position.

DOI
10.1201/9780429329050-2

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The concept of a Madelung fluid makes us accept that the matter per se admits at least a description, if not even a physical structure, by ensembles of free Hertz material particles. This chapter elucidates the idea of physical freedom, according to the nonstationary Schrödinger equation. Louis de Broglie observed that either the standard deviation in the case of Gaussian, or the period ‘t’ in the case of Airy function, should be taken as ‘uncertainty parameters’ rather than geometrical quantities. According to Michael Berry and Nandor Balazs, the argument of Airy function actually represents a caustic, but in the phase space: it is the envelope of the ensemble of straight lines representing the corresponding uniform motions. The Berry-Klein theory of scaling the forces, in case it can be applied to a Wien-Lummer enclosure, acts as a universal gauging procedure which, in an undertaking of Georgescu-Roegen type, should be able to make us decide what is infrafinite, finite or transfinite.

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Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position-momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

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Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations that is semi classical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum fluid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity field as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential was externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed fluids have the main application of this equation. Quantum fluids show strange behaviour when compared to normal fluids. It is also shown that quantum fluid also has spins which has no classical analog.

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After the fundamental works of Erwin Schrödinger, Erwin Madelung came up with a physical interpretation of the wave function, from the point of view of the classical fluids’ dynamics. This chapter is concerned with Madelung’s physical interpretation. The Madelung fluid is one case illustrating how “the forces detemine the velocities”, however at a different level of statistics. The “complex phase” of the wave function is the mark of the solution of nonstationary equation. However, his justification for the existence of the essentially complex wave function can be recognized only as anecdotal. The nonstationary Schrödinger equation admits, besides the classical Galilei group proper, an extra set of symmetries that can be taken in a form involving just one space dimension and time. The Maxwellian form of electrodynamics shows that, as long as the time is classically considered i.e. as just a parameter of continuity this concept of local does not require any further consideration beyond in a given space position.
The concept of a Madelung fluid makes us accept that the matter per se admits at least a description, if not even a physical structure, by ensembles of free Hertz material particles. This chapter elucidates the idea of physical freedom, according to the nonstationary Schrödinger equation. Louis de Broglie observed that either the standard deviation in the case of Gaussian, or the period ‘t’ in the case of Airy function, should be taken as ‘uncertainty parameters’ rather than geometrical quantities. According to Michael Berry and Nandor Balazs, the argument of Airy function actually represents a caustic, but in the phase space: it is the envelope of the ensemble of straight lines representing the corresponding uniform motions. The Berry-Klein theory of scaling the forces, in case it can be applied to a Wien-Lummer enclosure, acts as a universal gauging procedure which, in an undertaking of Georgescu-Roegen type, should be able to make us decide what is infrafinite, finite or transfinite.
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position-momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.
Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations that is semi classical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum fluid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity field as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential was externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed fluids have the main application of this equation. Quantum fluids show strange behaviour when compared to normal fluids. It is also shown that quantum fluid also has spins which has no classical analog.
Abstract Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations, that is semiclassical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum fluid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity field as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential were externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed fluids have the main application of this equation. Quantum fluids show strange behaviour when compared to normal fluids. It is also shown that quantum fluid also have spins which has no classical analog.
We construct a Madelung fluid model with specific time variable parameters as dissipative quantum fluid and linearize it in terms of Schrodinger equation with time dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrodinger equation and the corresponding classical linear ODE with variable frequency and damping. For the complex velocity field the Madelung system takes the form of a nonlinear complex Schrodinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we discuss and give exact results for nonlinear Madelung systems related with Caldirola-Kanai type dissipative harmonic oscillator.
Abstract We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.