Quantum Potential Research Articles

Interactions of ion acoustic multi-soliton and rogue wave with Bohm quantum potential in degenerate plasma

This work investigates the interactions among solitons and their consequences in the production of rogue waves in an unmagnetized plasmas composing non-relativistic as well as relativistic degenerate electrons and positrons, and inertial non-relativistic helium ions. The extended Poincarée–Lighthill–Kuo (PLK) method is employed to derive the two-sided Korteweg–de Vries (KdV) equations with their corresponding phase shifts. The nonlinear Schrödinger equation (NLSE) is obtained from the modified KdV (mKdV) equation, which allows one to study the properties of the rogue waves. It is found that the Fermi temperature and quantum mechanical effects become pronounced due to the quantum diffraction of electrons and positrons in the plasmas. The densities and temperatures of the helium ions, degenerate electrons and positrons, and quantum parameters strongly modify the electrostatic ion acoustic resonances and their corresponding phase shifts due to the interactions among solitons and produce rogue waves in the plasma.

Using Empirical Data to Estimate Potential Functions in Commodity Markets: Some Initial Results

This paper focuses on estimating real and quantum potentials from financial commodities. The log returns of six common commodities are considered. We find that some phenomena, such as the vertical potential walls and the time scale issue of the variation on returns, also exists in commodity markets. By comparing the quantum and classical potentials, we attempt to demonstrate that the information within these two types of potentials is different. We believe this empirical result is consistent with the theoretical assumption that quantum potentials (when embedded into social science contexts) may contain some social cognitive or market psychological information, while classical potentials mainly reflect ‘hard’ market conditions. We also compare the two potential forces and explore their relationship by simply estimating the Pearson correlation between them. The Medium or weak interaction effect may indicate that the cognitive system among traders may be affected by those ‘hard’ market conditions.

Some Clarifications on the Relation Between Bohmian Quantum Potential and Mach’s Principle

Mach's principle asserts that the inertial mass of a body is related to the distribution of other distant bodies. This means that in the absence of other bodies, a single body has no mass. In this case, talking about motion is not possible, because the detection of motion is possible only relative to other bodies. But in physics we are faced with situations that are not fully Machian. As in the case of general theory of relativity where geodesics exist in the absence of any matter, the motion has meaning. Another example which is the main topic of our discussion, refers to Bohmian quantum mechanics, where the inertial mass of a single particle does not vanish, but is modified. We can call such situations in which motion or mass of a single particle has meaning, pseudo-Machian situations. In this paper, we use the Machian or pseudo-Machian considerations to clarify under what circumstances and how a Machian effect leads us to Bohmian quantum mechanics. Then, we shall get the Bohmian quantum potential and its higher order terms for the Klein-Gordon particle through Machian considerations, without using any quantum mechanical postulate or operator formalism.

Covariant quantizations in plane and curved spaces

We present covariant quantization rules for nonsingular finite-dimensional classical theories with flat and curved configuration spaces. In the beginning, we construct a family of covariant quantizations in flat spaces and Cartesian coordinates. This family is parametrized by a function omega (theta ), theta in (1,0), which describes an ambiguity of the quantization. We generalize this construction presenting covariant quantizations of theories with flat configuration spaces but already with arbitrary curvilinear coordinates. Then we construct a so-called minimal family of covariant quantizations for theories with curved configuration spaces. This family of quantizations is parametrized by the same function omega (theta ). Finally, we describe a more wide family of covariant quantizations in curved spaces. This family is already parametrized by two functions, the previous one omega (theta ) and by an additional function varTheta (x,xi ). The above mentioned minimal family is a part at varTheta =1 of the wide family of quantizations. We study constructed quantizations in detail, proving their consistency and covariance. As a physical application, we consider a quantization of a non-relativistic particle moving in a curved space, discussing the problem of a quantum potential. Applying the covariant quantizations in flat spaces to an old problem of constructing quantum Hamiltonian in polar coordinates, we directly obtain a correct result.

Brownian motion of a classical particle in quantum environment

The Klein–Kramers equation, governing the Brownian motion of a classical particle in a quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.

Equivalent Quantum Equations in a System Inspired by Bouncing Droplets Experiments

In this paper we study a classical and theoretical system which consists of an elastic medium carrying transverse waves and one point-like high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this system. Afterwards we always consider the case where the concretion is not the wave source any longer. Then the concretion obeys a general and covariant guidance formula, which leads in low-velocity approximation to an equivalent de Broglie-Bohm guidance formula. The concretion moves then as if exists an equivalent quantum potential. A strictly equivalent free Schr\"odinger equation is retrieved, as well as the quantum stationary states in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation is fulfilled, it strikingly appears that the energy and momentum of the concretion not only are written in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale’s theory of scale relativity relying on a fractal space-time, we derive a generalized Schrodinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale’s theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrodinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrodinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrodinger equation satisfies an H-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrodinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

Halogen Bonds Involved in Copper(I) Complexes: A Study Based on the Electronic Charge Density

This communication describes the crystal structures of CuI complexes and their topological analysis with an emphasis on the Laplacian of the electron density to investigate the characteristics of halogen bonding. To gain insight into the halogen bonds (XBs), we survey wavefunction and DFT methods. The different XBs, that is, Cl···Cl–, I···I–, Br···N3–, and I···SCN–, in the crystal packing of these compounds are categorized as a combination of a region of charge depletion and a region of charge concentration in the valence‐shell charge concentration or hole–lump interactions. The full quantum potential based lump–hole concept is more useful than the σ‐hole concept, in which the electrostatic portion of the potential is merely considered. Such a view of halogen bonding can rationalize the geometry around the XBs. The noncovalent interaction reduced density gradient (NCI‐RDG) approach was applied to the real‐space visualization and quantitative investigation of the XBs as well.

Perturbative instability of inflationary cosmology from quantum potentials

It was argued that the Raychaudhuri equation with a quantum correction term seems to avoid the Big Bang singularity and to characterize an everlasting Universe [PLB741,276(2015)]. Critical comments on both conclusions and on the correctness of the key expressions of this work were discussed in literature [MPLA31(2016)1650044]. In the present work, we have analyzed the perturbative (in)stability conditions in the inflationary era of the early Universe. We conclude that both unstable and stable modes are incompatible with the corresponding ones obtained in the standard FLRW Universe. We have shown that unstable modes do exist at small (an)isotropic perturbation and for different equations of state. Inequalities for both unstable and stable solutions with the standard FLRW space were derived. They reveal that in the FLRW flat Universe both perturbative instability and stability are likely. While negative stability modes have been obtained for radiation- and matter-dominated eras, merely, instability modes exist in case of a finite cosmological constant and also if the vacuum energy dominates the cosmic background geometry.

Partial hydrodynamic representation of quantum molecular dynamics.

A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrödinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.

Quantum geometry of resurgent perturbative/nonperturbative relations

For a wide variety of quantum potentials, including the textbook ‘instanton’ examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain mathcal{N} = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and ‘special geometry’. These systems inherit a natural modular structure corresponding to Ramanujan’s theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

Quantum clustering in non-spherical data distributions: Finding a suitable number of clusters

Quantum Clustering (QC) provides an alternative approach to clustering algorithms, several of which are based on geometric relationships between data points. Instead, QC makes use of quantum mechanics concepts to find structures (clusters) in data sets by finding the minima of a quantum potential. The starting point of QC is a Parzen estimator with a fixed length scale, which significantly affects the final cluster allocation. This dependence on an adjustable parameter is common to other methods. We propose a framework to find suitable values of the length parameter σ by optimising twin measures of cluster separation and consistency for a given cluster number. This is an extension of the Separation and Concordance framework previously introduced for K-means clustering. Experimental results on two synthetic data sets and three challenging real-world data sets show that optimisation of cluster separation identifies QC solutions with consistently high Jaccard score measured against true-cluster labels while optimisation of cluster consistency provides insights into hierarchical cluster structure.

Quantum potential in covariant quantum mechanics

We discuss several features of the classical quantum potential appearing in Covariant Quantum Mechanics.In particular, we compare the “observed potential” A[K,G,o] of the joined spacetime connection K with the potential A↑ of the cosymplectic phase 2-form Ω[K,G] and with the potential A↑ of the upper quantum connection Ч↑.Moreover, we discuss the distinguished observer o[Ψ] and the distinguished timelike potential A[Ψ] associated with a non-vanishing quantum section Ψ.We show that the above objects play a natural role in the context of the kinetic quantum momentum Q[Ψ], of the quantum probability current J[Ψ], of the Schrödinger operator S[Ψ] and of the classical fluid associated with a non-vanishing quantum section Ψ.

The Dirac–Hestenes Equation and its Relation with the Relativistic de Broglie–Bohm Theory

In this paper we provide using the Clifford and spin-Clifford formalism and some few results of the extensor calculus a derivation of the conservation laws that follow directly from the Dirac–Hestenes equation (DHE) describing a Dirac–Hestenes spinor field (DHSF) in interaction with an external electromagnetic field without using the Lagrangian formalism. In particular, we show that the energy-momentum and total angular momentum extensors of a DHSF is not conserved in spacetime regions permitting the existence of a null electromagnetic field F but a non null electromagnetic potential \(A \). These results have been used together with some others recently obtained (e.g., that the classical relativistic Hamilton–Jacobi equation is equivalent to a DHE satisfied by a particular class of DHSF) to obtain the correct relativistic quantum potential when the Dirac theory is interpreted as a de Broglie–Bohm theory. Some results appearing in the literature on this issue are criticized and the origin of some misconceptions is detailed with a rigorous mathematical analysis.

Asymptotic stability of the rarefaction wave for the compressible quantum Navier–Stokes–Poisson equations

In this study, we consider the large time behavior of the solution to the one-dimensional isentropic compressible quantum Navier–Stokes–Poisson equations. The system describes a compressible particle fluid under quantum effects with the potential function of the self-consistent electric field. We show that if the initial data are close to a constant state with asymptotic values at far fields selected such that the Riemann problem on the corresponding Euler system admits a rarefaction wave with a strength that is not necessarily small, then the solution exists for all time and it tends to the rarefaction wave as t→+∞. The proof is based on the energy method by considering the effect of the self-consistent electric field and quantum potential in the viscous compressible fluid. In addition, we compare the quantum compressible Navier–Stokes–Poisson equations and the corresponding compressible Navier–Stokes–Poisson equations based on the large-time behavior of these two classes of models.

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in Cartesian coordinates. A few particular cases of these types of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of , and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the limit.

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrodinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrodinger equation may be called stochastic Schrodinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrodinger equation reduces to normal Schrodinger equation. It is found that the Schrodinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrodinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrodinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.

Blow-up of solutions to quantum hydrodynamic models in half space

In this paper, we prove that any smooth solutions of quantum hydrodynamic model which satisfies suitable conditions will blow up in finite time in half space. This model can be considered as the compressible Euler equation with quantum potential. The main ideal is based on the construction of energy inequality.

Analytic Solutions of the Madelung Equation

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.

The Planck Law for Particle with Rest Mass

In this work the author derives the Plank law for particles with rest mass that in the limit of photons with null rest mass leads to the known Plank law for the black body radiation. The paper shows that the relation has a pure quantum origin deriving it from the quantum potential of the hydrodynamic model. The paper puts under light how these laws originates by the scale invariance breaking due to the quantum potential. The paper also briefly analyzes the consequences of this symmetry breaking on the small-size black holes formation. PACS: 42.50.Lc, 04.60.-m