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arXiv:0704.2339

Scale invariant quantum potential leading to globally self-trapped wave function in Madelung fluid

Agus Budiyono

Published Apr 18, 2007• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0704.2339

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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.
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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 In single-particle Madelung mechanics, the single-particle quantum state Ψ ( x → , t ) = R ( x → , t ) e i S ( x → , t ) / ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R ( x → , t ) 2 and local momentum ∇ → S ( x → , t ) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q ( x → , t ) = − ℏ 2 2 m ∇ → R ( x → , t ) R ( x → , t ) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q ( x → , t ) < 0 . For energy superposition states with band-limit
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.