On geometro-stochastic theory of measurement and quantum geometry

The reasons for the fundamental incompatibility of quantum mechanics with classical relativistic geometries are reviewed, whereupon the basic principles of a theory of measurement leading to quantum geometries are stated and discussed. The ensuing conceptualization of quantum processes is formulated as an integral part of an all-pervasive concept of quantum reality in which systems as well as apparatuses are treated as quantum objects. The basic ideas of the resulting geometro-stochastic theory of quantum measurement are explained.     

Ingoing and outgoing waves in the non-relativistic theory of photoionization

A well-known formula for the differential cross section for photoionization contains a so-called ‘final state’ wave function which asymptotically behaves as a Coulomb-modified plane wave plus an ingoing Coulomb-modified radial wave. We explain the reason of appearance of that function on the ground of an analysis of an asymptotic form of a relevant outgoing Green's function. The reasoning is carried out without referring to an intuitive argumentation of Breit and Bethe [Phys. Rev. 93 (1954) 888].     

Some aspects of gravitational quantum mechanics

String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and free particle wave function. In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.     

On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

In this Letter, the generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. A generalized Darboux transformation (DT) of the GNLS equation is constructed with the help of the gauge transformation for an Ablowitz–Kaup–Newell–Segur (AKNS) type GNLS spectral problem, from which a unified formula of Nth-order rogue wave solution to the GNLS equation is given. In particular, the first and second-order rogue wave solutions to the GNLS equation are explicitly illustrated through some figures.     

On the existence of hydrogen atoms in higher dimensional Euclidean spaces

The question of whether hydrogen atoms can exist or not in spaces with a number of dimensions (D) greater than 3 is revisited. The lowest quantum mechanical stable states and the corresponding wave functions are determined by applying Numerov?s method to solve Schrödinger?s equation. States for different angular momentum quantum number and dimensionality are considered. One is lead to the result that hydrogen atoms in higher dimensions could actually exist. The most probable distance between the electron and the nucleus are then computed as a function of D showing the possibility of tiny confined states.     

A Tentative Expression of the Károlyházy Uncertainty of the Space-Time Structure Through Vacuum Spreads in Quantum Gravity

In the existing expositions of the Károlyházy model, quantum mechanical uncertainties are mimicked by classical spreads. It is shown how to express those uncertainties through entities of the future unified theory of general relativity and quantum theory.     

A Recurrent Quantum Neural Network Model to Describe Eye Tracking of Moving Targets

A theoretical quantum neural network model is proposed using a nonlinear Schrödinger wave equation. The model proposes that there exists a nonlinear Schrödinger wave equation that mediates the collective response of a neural lattice. The model is used to explain eye movements when tracking moving targets. Using a recurrent quantum neural network(RQNN) while simulating the eye tracking model, two very interesting phenomena are observed. First, as eye sensor data is processed in a classical neural network, a wave packet is triggered in the quantum neural network.This wave packet moves like a particle. Second, when the eye tracks a fixed target, this wave packet moves not in a continuous but rather in a discrete mode. This result reminds one of the saccadic movements of the eye consisting of ‘jumps’ and ‘rests’. However, such a saccadic movement is intertwined with smooth pursuit movements when the eye has to track a dynamic trajectory. In a sense, this is the first theoretical model explaining the experimental observation reported concerning eye movements in a static scene situation. The resulting prediction is found to be very precise and efficient in comparison to classical objective modeling schemes such as the Kalman filter.     

On the derivation of the Schrödinger equation from stochastic mechanics

It is shown that the existing formulations of stochastic mechanics are not equivalent to the Schrödinger equation, as had previously been believed. It is argued that this is a reflection of fundamental inadequacies in the physical foundations of stochastic mechanics.Some relatively minor difficulties with the demonstration of equivalence are already known for the special case in which the nodal surface separates the manifold of the diffusion into disjoint components.^(1,11) The problems described in this paper are much more general and quite unrelated.     

On the Number of Negative Eigenvalues of a One-Dimensional Schrödinger Operator with Point Interactions

An effective algorithm is provided for determining the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions in terms of the intensities and the distances between the interactions.     

Analyticity of the density of states in the anderson model on the Bethe lattice

Let H=1/2+V on l²(B), whereB is the Bethe lattice andV(x),x B, are i.i.d.r.v.'s with common probability distribution. It is shown that for distributions sufficiently close to the Cauchy distribution, the density of states(E) is analytic in a strip about the real axis.     

The Band-Edge Behavior of the Density of Surfacic States

This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered: fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably 'reduced' to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.Mathematics Subject Classifications (2000) 35P20, 46N50, 47B80.     

The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμ_λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm-Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ_m,n and the energies E_m,n of the bound states are exactly obtained in both the sphere S² and the hyperbolic plane H².     

On the Structure of Singularities of Integrable Schrödinger Operators

We prove that the singularities of the potential of a (multidimensional) Schrödinger operator obtained from the standard Laplacian by a Darboux transformation are located on a union of hyperplanes.     

金属/掺杂聚合物/金属结构中载流子的注入与输运

基于紧束缚的Su-Schrieffer-Heeger模型，利用非绝热的动力学方法，研究了金属/聚合物/金属三明治结构中载流子注入与输运的动力学过程，其中，聚合物中含有一个掺杂离子. 电子波函数随时间的演化满足含时Schrdinger方程，晶格运动满足经典的牛顿运动方程. 研究发现，掺杂离子对注入到聚合物中的载流子的作用类似一个控制阀门，阀门的状态与电场的强度和偏压的大小密切相关.     

金属/掺杂聚合物/金属结构中载流子的注入与输运

基于紧束缚的Su-Schrieffer-Heeger模型,利用非绝热的动力学方法,研究了金属/聚合物/金属三明治结构中载流子注入与输运的动力学过程,其中,聚合物中含有一个掺杂离子. 电子波函数随时间的演化满足含时Schrdinger方程,晶格运动满足经典的牛顿运动方程. 研究发现,掺杂离子对注入到聚合物中的载流子的作用类似一个控制阀门,阀门的状态与电场的强度和偏压的大小密切相关. 关键词： 载流子输运 波包 非绝热动力学     

On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.