Reply to Comments of Steuernagel on the Afshar’s Experiment

We respond to criticism of our paper “Paradox in Wave-Particle Duality for Non-Perturbative Measurements”. We disagree with Steuernagel’s derivation of the visibility of the Afshar experiment. To calculate the fringe visibility, Steuernagel utilizes two different experimental situations, i.e. the wire grid in the pattern minima and in the pattern maxima. In our assessment, this procedure cannot lead to the correct result for the complementarity properties of a wave-particle in one particular experimental set-up.     

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L ¹-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L ¹-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L ¹-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L ¹-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates. E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported by POCI/MAT/61931/2004. X. Lu work partially supported by NSF of China grant 10571101.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L ^∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and L ^p estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions. Supported by the Scientific Research Foundation of Huazhong University of Science and Technology (HUST-SRF).     

Reply to the comment on “On the Barut-Vigier model of the hydrogen atom”

It is shown that the main conclusion of Dragić and Marić [Phys. Lett. A 229 (1997) 130] about the ground state of the hydrogen atom in the Barut-Vigier potential is based on the wrong use of the radial wave function which cannot be normalized.     

Comments on “Barut-Girardello Coherent States for the Parabolic Cylinder Functions”

In Chenaghlou and Faizy (Int. J. Theor. Phys. 2008), the authors claim that they have constructed the Barut-Girardello coherent states for the parabolic cylinder functions. However, we point out here that by introducing these coherent states, Schrödinger was able to put forth the idea of “coherent states of the quantum harmonic oscillator” over eighty years ago. These coherent states are derived not only from the Barut-Girardello eigenvalue equation, but also from the Schrödinger and the Klauder-Perelomov approaches. Thus, contrary to their claim, the authors have not introduced new coherent states. In particular, a wide range of the parabolic cylinder functions do not form an orthonormal basis.     

The Solutions for the Boundary Layer Problem of Boltzmann Equation in a Half-Space

We study the half space boundary layer problem for Boltzmann equation with cut-off potentials in all the cases −3<γ≤1, while the boundary condition is imposed on the incoming particles of Dirichlet type, and the solution is assumed to approach to a global Maxwellian at the far field. The same as for cut-off hard sphere model, there is an implicit solvability condition on the boundary data which gives the co-dimensions of the boundary data in terms of positive characteristic speeds.     

Comments on “There Is No Axiomatic System for the Quantum Theory”

In a recent paper, Nagata (Int. J. Theor. Phys. 48(12):3532, 2009) claims to derive inconsistencies from quantum mechanics. In this paper, we show that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables.     

Comments on counterexamples to the cosmological “no hair” conjecture

It is shown that two recently published counterexamples to the cosmological “no hair” conjecture are merely coordinate transforms of the de Sitter spacetime.     

Bondi-Metzner-Sachs Symmetry,Holography on Null-surfaces and Area Proportionality of “Light-slice” Entropy

It is shown that certain kinds of behavior, which hitherto were expected to be characteristic for classical gravity and quantum field theory in curved spacetime, as the infinite dimensional Bondi-Metzner-Sachs symmetry, holography on event horizons and an area proportionality of entropy, have in fact an unnoticed presence in Minkowski QFT.     

Reply to “Comments on Bouda and Djama's ‘Quantum Newton's Law’”

In this reply, we hope to bring clarifications about the reservations expressed by Floyd in his comments, give further explanations about the choice of the approach and show that our fundamental result can be reproduced by other ways. We also establish that Floyd's trajectories manifest some ambiguities related to the mathematical choice of the couple of solutions of Schrödinger's equation.     

Solution to the Schrödinger Equation for the Time-Dependent Potential

In this work, the Schrödinger equation with the time-dependent potential \(V(z,\hat{p},t)=g_{1}(t)z+g_{2}(t)\hat{p}+g_{3}(t)\) has been solved by the method of time-space transformation in 1+1 dimensions. The corresponding analytical wave function to Schrödinger equation is obtained. In addition, the discussion of solutions to particular cases has been made.     

pH Dependence of the Au-CN Species Adsorbed onto Activated Carbon — Reply to Comments

Several aspects of a recent paper [McGrath, A. C. et al., Hyp. Interact. 139/40 (2002), 471] have been criticised by Klauber [Klauber, C., Hyp. Interact. 155 (2004), 65]. Although some of the points in Klauber [Klauber, C., Hyp. Interact. 155 (2004), 65] are shown to be incorrect, we show that a further analysis of the data in McGrath et al. [McGrath, A. C. et al., Hyp. Interact. 139/40 (2002), 471] does lend support to Klauber's tetramer model for the AuCN species on activated carbon, but not necessarily for its method of bonding.     

Reply to “Comments on ‘How Does a Crossed-Coil NMR Spectrometer Work?’”

I am delighted that my recent article in this journal¹ on the operation of a crossed-coil NMR spectrometer has stimulated a discussion² of this intriguing problem, and appreciate this opportunity to reply.     

On Unique Symmetry of Two Nonlinear Generalizations of the Schrödinger Equation

Abstract

We prove that two nonlinear generalizations of the nonlinear Schrödinger equation are invariant with respect to a Lie algebra that coincides with the invariance algebra of the Hamilton-Jacobi equation.