Quantum Mechanical Equations Research Articles

ПОЛУЧЕНИЕ МАКРОМЕХАНИЧЕСКИХ ВЕЛИЧИН С ПОМОЩЬЮ КВАНТОВО-МЕХАНИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

The wave function Ψ satisfies the Schrödinger equation for a free particle. Formally, the Schrödinger equation generates the magnitude of mechanical motion of the zero order 0 p = mv 0 (in the sense that it is contained in the Schrödinger equation. Comparison of the wave function Ψ and its gradient implies a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the first order 1 p = mv 1. From the comparison of the wave function and its time derivative yields a formal analogue of the Schrödinger equation, which generates the magnitude of mechanical motion of the second order 2 p = mv 2/2!. The values of mechanical motion of the zero, first, and second orders are known.Obviously, other formal analogs of the Schrödinger equation can generate mechanical motion of other orders.The aim of the work is to establish such quantities and related regularities that may be of interest, which makes the study relevant. hovering. Then the spatial derivatives will be one-dimensional. The magnitude of the mechanical movement of the third order is 3 p = mv 3/3!. This value is Umov's integral vector for kinetic energy. The magnitude of the mechanical movement minus the first order -1 p = mv -1 is a reverse impulse. The meaning of this quantity and its relevance is established by the theorem: in a hy- drogen-like atom, the quantity m v-1 is quantized. A fixed (unchanged) quantum is a quantity m v-1 correspond- e e 0 ing to the basic energy level. Almost all of the results obtained were a consequence of the use of quantum mechanical differential equations, however, the results themselves are predominantly macromechanical. The quantities of mechanical motion of various orders are generated by formal analogs of the Schrödinger equation. In all formal analogs of the Schrödinger equation, the orders of the partial derivatives differ by one. For quantities of motion with a positive degree of velocity, the order of the temporal derivatives is higher than that of the spatial ones. For quantities with a negative degree, the order of spatial derivatives is higher.

The influence of resonator outcoupling and losses on the photon statistics of the LASER

An extension to the quantum-mechanical laser master equation for the density operator is derived to incorporate the beam-splitter effect caused by typical dielectric laser output couplers. This effect gives rise to a significant change in the photon statistical distribution of the part of the laser light reflected back into the resonator and, therefore, may have an influence on the total laser output photon statistics. Different cases without and with additional intra-cavity losses were discussed and their influence on the expected laser photon statistics was deduced. As a result, it was found that the well-known Poisson distribution of laser light is in most cases the result of additional losses or absorption, which act uncorrelatedly on single photons. In a laser with negligible additional losses where outcoupling is dominated by the beam-splitter effect, the photon statistics reveal to be mainly non-Poisson. A Poisson distribution would only occur for very low outcoupling rates, i.e., high finesse cavities. It is found that in the limit of strong outcoupling even the distribution of thermal light can result.

Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics

In this paper, we consider a coupled space-time fractional nonlinear Schrödinger type equation which can be used for describing nonrelativistic quantum mechanical behavior. Under the help of the fractional complex transform and the conformable fractional derivative sense, we apply the extended auxiliary equation method, the modified He’s semi-inverse method and the improved Weierstrass elliptic function method to obtain many exact solutions. These exact solutions include the bright, dark, bright-dark, bright-like and kinky-bright optical soliton solutions as well as trigonometric function solutions, the Jacobi elliptic function solutions and the general solutions in form of Weierstrass elliptic function. The behaviors of some obtained solutions are displayed through the 3D graphs for different fractional orders by using Matlab. The obtained results show that these proposed methods are straightforward, efficient and can provide more forms of exact solutions, which are helpful to understand the complex physical meaning of space-time fractional coupled nonlinear Schrödinger equation.

Clinical Application of Graphene Composite in Internal Fixation of Ankle Fracture in Sports

The ankle joint consists of the tibia, the fibrous lower end, and the talus. Osteoporotic fractures and deboning are common injuries in orthopaedic surgery, often due to abnormal disorders following an ankle fracture. Various fractures can occur depending on the shape, size, and position of the foot at the time of the injury. With the continuous development of science and the continuous improvement of modern sports level, scientists and sports workers worldwide recognize the importance of applying new materials in sports equipment. Composite material is a combination of multiphase combination material. In short, two or more components with different properties and different forms are used in a multimaterial combination of composite means. The organic combination of related scientific equipment and composite materials promotes the development of sports equipment and the probability of reducing sports injury. The function of the existing graphene composite is analyzed and applied, and the combination of graphene composite and sports is realized theoretically. The clinical application of graphene composite in sports is studied to promote sports development. This article mainly studies the clinical application of graphene composite in ankle fracture internal fixation in sports. It is found that with the increasingly mature application of graphene rubber composite, graphene fiber composite, and other graphene composite materials, it can be widely used in all aspects of sports equipment. Graphene composite materials can be used more easily in sports equipment. In this article, the thermal material conversion algorithm, Schrodinger equation in quantum mechanics, and the use method of graphene composite materials are used to study the clinical application of graphene composite in the internal fixation of ankle joint fracture in sports. The frequent injury and instability of ankle and knee joints are signs of ankle and knee joint injury. The main factors that affect the instability of ankle and knee function are the comprehensive effects of motion range, muscle force valgus, and body feeling, and graphene composite materials are widely used in the internal fixation of ankle joint fracture in sports. The results show that graphene composite can be used in sports equipment and has a great space and high feasibility through the analysis of four groups of sports equipment. It shows that advanced materials play a very important role in the research of sports equipment.

The modified fundamental equations of quantum mechanics

The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.

Liouvillian skin effect in an exactly solvable model

The interplay between dissipation, topology and sensitivity to boundary conditions has recently attracted tremendous amounts of attention at the level of effective non-Hermitian descriptions. Here we exactly solve a quantum mechanical Lindblad master equation describing a dissipative topological Su-Schrieffer-Heeger (SSH) chain of fermions for both open boundary condition (OBC) and periodic boundary condition (PBC). We find that the extreme sensitivity on the boundary conditions associated with the non-Hermitian skin effect is directly reflected in the rapidities governing the time evolution of the density matrix giving rise to a Liouvillian skin effect. This leads to several intriguing phenomena including boundary sensitive damping behavior, steady state currents in finite periodic systems, and diverging relaxation times in the limit of large systems. We illuminate how the role of topology in these systems differs in the effective non-Hermitian Hamiltonian limit and the full master equation framework.

Exact quantum-mechanical equations for particle beams

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

The Cosmic Black Hole as a Solution of the Relativistic Quantum Mechanical DIRAC Equation

For many physicists, Albert Einstein’s Theory of General Relativity is the top of Physics. A whole new concept, based on a flexible Space-Time Continuum. A wonderful New and Original insight in the Origins of Space and Time. But nowadays physics requires more than a fundamental theory about Space and Time.
 The Mathematical foundation for a “Quantum Mechanical Model of the Black Hole” is based on a 10-dimensional Space-Time Continuum. This article has been written in projections of a 10-Dimensional Space-Time Continuum within an easier to understand 4-Dimensional Space-Time Continuum. For that reason, this theory will not start with “Einstein’s famous Field Equations”, but the start will be at a very fundamental concept in Physics. Isaac Newton’s 3rd law as a fundament in Classical- and Quantum Mechanics.
 To make the theory of the “Quantum Mechanical Model of the Black Hole” as much understandable as possible, this article starts with a short comprehension of the theory.

Newton’s Theory of Light and Wave-Particle Duality

The long history of the dispute among physicists as to whether matter is composed of particles or waves is reviewed. A turning point came at the dawn of the 20th century in the form of the de Broglie momentum-wavelength and Bohr/Planck energy frequency relations. It was on this basis that Schrödinger developed a quantum mechanical differential equation whose solution came to be known as a wave function. The Born Interpretation of these functions holds that their absolute square constitutes a probability distribution which has been used successfully to make predictions of the properties of the system under discussion. It is pointed out that the phenomenon of light refraction played a key role in the development of both Newton’s corpuscular theory of light and the competing theory of Huygens which looks upon light as consisting exclusively of waves. It is shown that Newton’s failure to predict the decrease in the speed of light as it passes from air into water was not because of his belief in the particle composition of light, but rather because he not did anticipate that the mass of a photon changes upon entering a medium of higher index of refraction n. By assuming that the energy E of light is equal pc/n, where p is the momentum of the photons and c is the speed of light in free space, it is shown that the experimental dependence of the speed the light on its wavelength as it passes through a transparent medium is derived successfully through the use of Hamilton’s Canonical Equations and Newton’s Second Law of Kinetics. It is suggested that the relationship between particles and waves can be understood by noting that the localized properties of matter exhibit themselves in experiments such as the photoelectric effect where attention can be concentrated on the behavior of a single particle. The corresponding wave properties occur when large numbers of particles are observed under exactly similar circumstances, as for example in electron diffraction. ...........

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

Classical dynamics from self-consistency equations in quantum mechanics

During the last three decades, Pavel Bóna developed a non-linear generalization of quantum mechanics, which is based on symplectic structures for normal states. One important application of such a generalization is a general setting that is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bóna’s non-linear quantum mechanics. It is based on C0-semigroup theory and has a domain of applicability that is much broader than Bóna’s original one. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. In this new mathematical approach, we build a Poisson bracket for the polynomial functions on the Hermitian weak*-continuous functionals on any C*-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C*-algebra by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C*-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C0-groups of contractions. As a matter of fact, in generic commutative C*-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak* Gâteaux derivative and the state-dependent C*-dynamical systems. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here.

Joint Natural and Fourier Transform Technique for Unified Fractional Schrodinger Equation in Quantum Mechanics

Integral transform method plays a significant role to solve different kinds of integral and differential equations of fractional order and integer order that arise in several applications of engineering and physical sciences. Natural transform converges to Laplace and Sumudu transforms. The aim of this present paper is to solve the unified fractional Schrödinger equation that occurs in the field of quantum mechanics. The solution is obtained by Natural transform technique for the Caputo fractional derivatives with time variable and Fourier transform for the Liouville fractional derivative with space variable. The result is provided in the computational form of the H-function. The paper explicitly reveals the efficiency and reliability of joint Natural and Fourier transform technique. Some special cases of the main result are mentioned. Achieved result is general in nature and has the capability of providing a number of known and new results.

Differential equations of quantum mechanics

We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.

A Machian Reformulation of Quantum Mechanics

The widely known but also somewhat esoteric Mach principle envisages a fully relational formulation of physical theories without any reference to a concept of ‘absolute space’. When applied to classical mechanics, under the guise of an extended symmetry group, this procedure is known to lead to an equation of motion with inertial-like forces that are sourced by the mass distribution of the system itself. In this paper we follow a similar procedure and reformulate the Schrödinger equation of non-relativistic quantum mechanics in a fully Machian way. Just like its classical counterpart, the resulting quantum theory is fully relational in the positions and momenta of the bodies comprising a given physical system, leaving no room for the notion of absolute space.

Signatures of atomic structure in subfemtosecond laser-driven electron dynamics in nanogaps

Coupled Maxwell and quantum-mechanical equations are used to simulate the electron dynamics in nanogaps in systems containing thousands of atoms. It will be shown that besides the carrier-envelop phase, bow-tie or gap shape, and gap size, the atomistic structure also significantly alters the electron dynamics. Atomic-scale interference fringes appear not only in the electron density but in the electron current and field enhancement as well. Electron bursts emerge from individual atoms and scatter on atoms driven by the direction of the laser. The time-dependent orbital-free density functional theory coupled to the Maxwell equations allows us to simulate physical systems approaching the realistic size and to explore the physical mechanism controlling the electron dynamics.

Reflection and refraction of photons

Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.

CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

Quantum mechanics: how to use Everett theory practically

Among theories of the physical world, quantum mechanics remains a topic of lively discussions on its so-called interpretation. For some it remains an open question to understand how deterministic equations of this theory, as established long ago, may combine with a fundamental uncertainty. We consider the process of spontaneous emission by an atom interacting with infinite number of degrees of freedom of the electromagnetic field. There is uncertainty in the evolution of the photo-emission process which was characterized as Markovian by using the equations of quantum mechanics when the decay of the atom is due to the coupling with the vacuum field. The Markovian property leads us naturally to describe spontaneous emission by using the classical Kolmogorov equation for the probability evolution of a parameter defining the state of the atom. We explain that Everett’s many-worlds interpretation weld together our description, and appears therefore as a consequence of the equations of quantum mechanics, that solves in this case the riddle of deterministic equations describing random events.

A New Form of Quantum Mechanical Wave Equation Unifying Quantum Mechanics and Electro-Magnetics

The Schr?dinger differential equation is what we usually solve for the microscopic particles in non-relativistic quantum mechanics. Niels Bohr suggested the power two of the (usually) complex answer shows the probability of the particle’s existence at a point of space. Also, the time dependence of Schrodinger wave equation is one whereas for light in electromagnetism is two. In this paper, we show a solution for both problems. We derive a Wave Equation for the energy of every system. This electromagnetic wave equation is shown to convert to those classical (i.e. the Schrodinger) and special relativistic (i.e. Klein-Gordon) quantum mechanical equations. Also, accordingly there definitely is a physical meaning to answer to this wave equation. And therefore, switching the probabilistic interpretation of quantum mechanics to a deterministic one as (Albert) Einstein demanded.

Estimation of Seismic Quality Factor via Quantum Mechanics-Based Signal Representation

We propose a stable seismic Q estimation approach by employing quantum mechanics-based signal representation. For Q estimation, we project a seismic trace onto a specific basis composed of wave functions constructed through the resolution of the Schroedinger equation of non-relativistic quantum mechanics at first. Then, based on the specific basis, we derive the quantum mechanics-based Q estimation approach in the local frequency-projection coefficient domain. The Planck constant and the control factor are the two key factors for the quantum mechanics-based Q estimation method. Compared with the traditional methods, the quantum mechanics-based Q estimation approach shows more stability and noise robustness. The synthetic and field data applications illustrate the effectiveness and the superiority of the proposed method. The quantum mechanics-based Q estimation method offers a new field and provides a complementary way for measuring seismic attenuation.