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arXiv:0902.3838 · Physical Review A

Madelung fluid model for the most likely wave function of a single free particle in a two-dimensional space with a given average energy

Agus Budiyono, Ken Umeno

Published Apr 23, 2009• 4 citations
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Abstract

We consider spatially two-dimensional Madelung fluid whose irrotational motion reduces into the Schr\"odinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is nonspinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schr\"odinger equation for a particle in a cylindrical tube external classical potential.

DOI
10.1103/physreva.79.042104

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We consider spatially two-dimensional Madelung fluid whose irrotational motion reduces into the Schr"odinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is nonspinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schr"odinger equation for a particle in a cylindrical tube external classical potential.
We show in spatially one dimensional Madelung fluid that a simple requirement on local stability of the maximum of quantum probability density will, if combined with the global scale invariance of quantum potential, lead to a class of quantum probability densities globally being self-trapped by their own self-generated quantum potentials, possessing only a finite-size spatial support. It turns out to belong to a class of the most probable wave function given its energy through the maximum entropy principle. We proceed to show that there is a limiting case in which the quantum probability density becomes the stationary-moving soliton-like solution of the Schrödinger equation.
We discuss two-dimensional Madelung fluid dynamics whose irrotational case reduces into the Schrödinger equation for a free single particle. We show that the self-trapped spinning-stationary Madelung fluid reported in the previous paper can be analogically identified as an equilibrium thermodynamics system. This is done by making correspondence between Shannon entropy over Madelung density and internal energy to be defined in the main text, respectively with thermal-entropy and thermal-internal energy of equilibrium thermodynamics system. This leads us to identify a Madelung fluid analog of thermal-temperature at the vanishing value of which the stationary Madelung fluid will be no more spinning and is equal to the quantum mechanical ground state of a particle trapped inside a cylindrical tube external potential.
Starting from relativistic mass-less Madelung fluid, we shall develop a class of typical wave functions by imposing it to maximize Shannon entropy given its finite average quantum potential. We show that there is a class of solutions in which the wave function is spatiotemporally localized with finite spacetime support, uniformly moving hence stationary. It turns out that the quantum amplitude satisfies Klein-Gordon equation with emergent mass term proportional to the square root of average quantum potential. We show that there is physical time uncertainty which decreases as the mass increases. We also rederive the classical energy-momentum relation provided the de Broglie-Einstein relation holds. In this case, the time uncertainty is proportional to the inverse of classical energy.
Abstract A formalism is developed, and applied, that describes a class of one-body quantum mechanical systems as fluids where each stationary state is a steady flow state. The time-independent Schrödinger equation for one-body stationary states with real-valued wavefunctions is shown to be equivalent to a compressible-flow generalization of the Bernoulli equation of fluid dynamics. The mass density, velocity and pressure are taken as functions that are determined by the probability density. The generalized Bernoulli equation describes compressible, irrotational, steady flow with variable mass and a constant specific total energy, i.e, a constant energy per mass for each fluid element. The generalized Bernoulli equation and a generalized continuity equation provide a fluid dynamic interpretation of a class of quantum mechanical stationary states that is an alternative to the unrealistic, static-fluid interpretation provided by the Madelung equations and quantum hydrodynamics. The total kinetic energy from the Bernoulli equation is shown to be equal to the expectation value of the kinetic energy, and the integrand of the expectation value of the kinetic energy is given an interpretation. It is also demonstrated that variable mass is necessary for a satisfactory fluid model of stationary states. However, over all space, the flows conserve mass, because the rate of mass creation from the sources are equal to the rate of mass annihilation from the sinks. The following flows are examined: the ground and first excited-states of a particle in a one-dimensional box, the harmonic oscillator, and the hydrogen s states.