Quantum Mechanics from the Energy Circulation Theory — Wave Function Showing an Energy Location in the 3D Real Space

The Schrödinger equation is one of the cores in quantum mechanics, but bears a contradiction. In the process to obtain the energy and momentum operators, the relation [Formula: see text] is used for [Formula: see text]. However, when they are applied to the Hamiltonian equation, the kinetic energy is set as [Formula: see text]. Based on the Energy Circulation Theory, we examine in this paper the quantization of motions of a particle. We clarify in which situation and for what energy we can use the relation [Formula: see text]. We derive a wave equation de novo to provide wave functions representing a concrete motion and energy distribution of a particle. The Schrödinger equation has a similar form by chance but the mass in our new equation is that of energy quantum expressed by [Formula: see text], which is common for any energies of any particles and decided only by the moving speed. A solution shows an energy location in the 3D real space even if it is expressed in complex. When a motion of a particle gets in circle, its circular frequency becomes quantized. In an atom, an electron circulates around the hidden dimensional axis, and the circulation can further rotate. We propose the quantization conditions for the electron orbiting, and derive the wave functions in concrete for S and P orbitals, which are different from current perceptions. We also demonstrate that the uncertainty principle is not valid for a motion of a particle.

Stability of a Point Charge for the Vlasov–Poisson System: The Radial Case

We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian equation in action-angle coordinates.

The Hamiltonian of the f(R) gravity

We derived conjugate momenta variables tensors and the Hamiltonian equation of the source free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables - fundamental metric tensor and its first ordinary partial derivatives with respect to spacetime coordinates and second ordinary partial derivatives with respect to spacetime coordinates - and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g. Einstein-Hilbert Lagrangian without matter fields) which is the same result obtained using the stress-energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any model of negative power law in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = α R^-r, where r and α are positive real numbers, it also forbids any polynomial equation which contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attains a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far reaching consequences as the result of applying the f(R) gravity to the study of Black Holes and the Friedmann-Lemaître-Robertson-Walker model in Cosmology.

Reformulation of Degasperis-Procesi Field by Functional Derivatives

We reformulated the Degasperis-Procesi equation using functional derivatives. More specifically, we used a semi-inverse method to derive the Lagrangian of the Degasperis-Procesi equation. After introducing the Hamiltonian formulation using functional derivatives, we applied this new formulation to the Degasperis-Procesi Equation. In addition, we found that both Euler-Lagrange equation and Hamiltonian equation yield the same result. Finally, we studied an example to elucidate the results.

Flexural Wave Propagation of Double-Layered Graphene Sheets Based on the Hamiltonian System

Double-layered graphene sheets (DLGSs) as a new type of nanocomponents, with special mechanical, electrical and chemical properties, have the potential of being applied in the nanoelectro-mechanical systems (NEMS) and nanoopto-mechanical systems (NOMS). In DLGSs structure, the two graphene sheets are connected by van der Waals (vdW) interaction. Thus, it can exhibit two vibration modes during the propagation of the flexural wave, i.e., in-phase mode and anti-phase mode. Based on the Kirchhoff plate theory and the nonlocal elasticity theory, Hamiltonian equations of the DLGSs are established by introducing the symplectic dual variables. By solving the Hamiltonian equation, the dispersion relation of the flexural wave propagation of the DLGSs is obtained. The numerical calculation indicates that the bending frequency, phase velocity and group velocity of the in-phase mode and anti-phase mode for the DLGSs are closely related to the nonlocal parameters, the foundation moduli and the vdW forces. The research results will provide theoretical basis for the dynamic design of DLGSs in micro-nanofunctional devices.

Study of Geometrically Necessary Dislocations of a Partially Recrystallized Aluminum Alloy Using 2D EBSD

During recrystallization, the growth of fresh grains initiated within a deformed microstructure causes dramatic changes in the dislocation structure and density of a heavily deformed matrix. In this paper, the microstructure of a cold rolled and partially recrystallized Al-Mg alloy (AA5052) was studied via electron backscattered diffraction (EBSD) analysis. The structure and density of the geometrically necessary dislocations (GNDs) were predicted using a combination of continuum mechanics and dislocation theory. Accordingly, the Nye dislocation tensor, which determines the GND structure, was estimated by calculation of the lattice curvature. To do so, five components of the Nye dislocation tensor were directly calculated from the local orientation of surface points of the specimen, which was determined by two-dimensional EBSD. The remaining components of GNDs were determined by minimizing a normalized Hamiltonian equation based on dislocation energy. The results show the elimination of low angle boundaries, lattice curvature, and GNDs in recrystallized regions and the formation of low angle boundaries with orientation discontinuities in deformed grains, which may be due to static recovery.

Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra

The QCD light-front Hamiltonian equation HLFΨ=M2Ψ derived from quantization at fixed LF time τ=t + z/c provides a causal, frame-independent method for computing hadron spectroscopy as well as dynamical observables such as structure functions, transverse momentum distributions, and distribution amplitudes. The QCD Lagrangian with zero quark mass has no explicit mass scale. de Alfaro, Fubini, and Furlan (dAFF) have made an important observation that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. If one applies the dAFF procedure to the QCD light-front Hamiltonian, it leads to a color-confining potential κ4ζ2 for mesons, where ζ2 is the LF radial variable conjugate to the qq¯ invariant mass squared. The same result, including spin terms, is obtained using light-front holography, the duality between light-front dynamics and AdS5, if one modifies the AdS5 action by the dilaton eκ2z2 in the fifth dimension z. When one generalizes this procedure using superconformal algebra, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The pion qq¯ eigenstate has zero mass at mq=0. The superconformal relations also can be extended to heavy-light quark mesons and baryons. This approach also leads to insights into the physics underlying hadronization at the amplitude level. I will also discuss the remarkable features of the Poincaré invariant, causal vacuum defined by light-front quantization and its impact on the interpretation of the cosmological constant. AdS/QCD also predicts the analytic form of the nonperturbative running coupling αs(Q2)∝e-Q2/4κ2. The mass scale κ underlying hadron masses can be connected to the parameter ΛMS¯ in the QCD running coupling by matching the nonperturbative dynamics to the perturbative QCD regime. The result is an effective coupling αs(Q2) defined at all momenta. One obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. Finally, I address the interesting question of whether the momentum sum rule is valid for nuclear structure functions.

A model of nonlinear DNA-protein interaction system with Cornell potential and its stability

<p class="Abstract">The purpose of this study was to determine the model of a interaction system between the DNA with protein. The interaction system consisted of a molecule of protein bound with a single chain of DNA. The interaction between DNA chain, especially adenine and thymine, and DNA-protein bound to glutamine and adenine. The forms of these bonds are adapted from the hydrogen bonds. The Cornell potential was used to describe both of the interactions. We proposed the Hamiltonian equation to describe the general model of interaction. Interaction system is divided into three parts. The interaction model is satisfied when a protein molecule triggers pulses on a DNA chain. An initial shift in position of protein xm should trigger the shift in position of DNA ym, or alter the state. However, an initial shift in DNA, yn, should not alter the state of a rest protein (i.e. xm = 0), otherwise, the protein would not steadily bind. We also investigated the stability of the model from the DNA-protein interaction with Lyapunov function. The stability of system can be determined when we obtained the equilibrium point.</p>

A model of nonlinear DNA-protein interaction system with Cornell potential and its stability

<p class="Abstract">The purpose of this study was to determine the model of a interaction system between the DNA with protein. The interaction system consisted of a molecule of protein bound with a single chain of DNA. The interaction between DNA chain, especially adenine and thymine, and DNA-protein bound to glutamine and adenine. The forms of these bonds are adapted from the hydrogen bonds. The Cornell potential was used to describe both of the interactions. We proposed the Hamiltonian equation to describe the general model of interaction. Interaction system is divided into three parts. The interaction model is satisfied when a protein molecule triggers pulses on a DNA chain. An initial shift in position of protein xm should trigger the shift in position of DNA ym, or alter the state. However, an initial shift in DNA, yn, should not alter the state of a rest protein (i.e. xm = 0), otherwise, the protein would not steadily bind. We also investigated the stability of the model from the DNA-protein interaction with Lyapunov function. The stability of system can be determined when we obtained the equilibrium point.</p>