Schrodinger Equation Research Articles

Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term

Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term

Optical Solitons for Nonlinear Schrodinger Equation Formatted in the Absence of Chromatic Dispersion Through Modified Exponential Rational Function Method and Other Distinct Schemes

Optical Solitons for Nonlinear Schrodinger Equation Formatted in the Absence of Chromatic Dispersion Through Modified Exponential Rational Function Method and Other Distinct Schemes

Algebraic Solution of the Quantum Harmonic Oscillator

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.

An analogy between effective potential representing prime numbers and Schwarzschild black hole

The effective two body interaction is derived using the single particle Schrödinger equation for the prime numbers with probability density based on the prime counting function (which is the number of primes below a certain x). Though the speculative connection of primes to Schwarzschild spacetime or black hole has been there for a long time, here for the first time our derived potential serendipitously shows the unique similarity with the effective potential derived from general theory of relativity (GTR) for Schwarzschild spacetime. Also, we report the existence of ISCO and photon sphere in prime potential and compare with GTR. It is found that when the prime numbers are represented by a quantum system, the two-body interactions mimic the effective general relativistic gravitational interaction in black hole with the four-acceleration diverging. We also establish an analogy between Einstein's static universe and the universe of primes. The question is: “Who ordered that?”

Control of fractional in-time Schrodinger equations via comprehensive Caputo derivative strategies

Control of fractional in-time Schrodinger equations via comprehensive Caputo derivative strategies

Travelling Wave-like Solutions for Some Nonlinear Equations with Cubic Nonlinearities

We exercised the series-expansion method to extract solitary wave solutions for complex nonlinear evolution equations (NLEEs) such as the coupled Higgs equation (CHE), the (3+1)-dimensional dynamical system, the (3+1)-dimensional nonlinear Schrodinger(NLSE) equation which have many physical importance in different branches. Varieties of periodic and solitonic wave-like solutions are extracted. Computational work has been done and plots and counter graphs are plotted using Wolfram Mathematica 11.

Propagation of spin waves in a lattice of laterally and vertically coupled YIG microwaveguides by changing the magnetization angle in linear and nonlinear modes

Purpose. Investigation of the joint manifestation of the effects of anisotropic signal propagation, coupling, and nonlinear power dependence of the medium parameters in a lattice of laterally and vertically coupled spin-wave (SW) microwaveguides. Consideration of the case of the influence of the rotation of the magnetization angle and the change of the lateral gap between microwaveguides located on the same substrate on the transverse profile of the spin-wave beam and the spatial localization of the SW amplitude. Methods. The method of micromagnetic modeling based on the numerical solution of the Landau–Lifshitz–Hilbert equation shows the possibility of controlling the direction of propagation of SW in an ensemble of laterally and vertically coupled iron yttrium garnet (YIG) microwaveguides by changing the magnetization angle. By the method of numerical integration of the system of coupled discrete nonlinear Schrodinger equations, the possibility of changing the transverse profile of the spin-wave beam by changing the level of the initial signal amplitude is shown. Results. The spatial distributions of the components of the dynamic magnetization of the SW excited in two microwaveguides located on the same substrate obtained in micromagnetic simulations indicate a change in the character of localization of the SW power in the output sections of the microwaveguides. At variation of the lattice magnetization angle, a shift of the threshold power value is observed, at which a characteristic curbing of the transverse width of the spin-wave beam in the nonlinear mode appears. Conclusion. When excitation of surface magnetostatic SW in a lattice of laterally and vertically coupled microwaveguides, a transformation of the transverse profile of the wave is observed at a deviation of the magnetization angle of the structure by 15º , which is manifested in the change of the SW length and its localization in each of the microwaveguides. The combined effects of dipole coupling, gyrotropy, and nonlinearity of the medium make it possible to control the value of the threshold power of the SW, at which the mode of diffractionless propagation of the spin-wave beam is realized in a single layer of the structure.

Effects of external magnetic fields on the confined states and optical properties of conical quantum dot nanowire

In this paper, we have investigated theoretically the effect of the external magnetic fields on the electronic structuctures and optical properties of a nanosystem consisting of a conical InAs quantum dot embedded in an infinite GaAs nanowire. To this end the Schrodinger equation have been solved numerically utilizing the effective mass approximation to calculate the confined energy states and binding energies of the system. The findings demonstrate that under the effect of the magnetic fields, the energies of confined states in the Quantum Dot of +|m| and −|m| angular momentum have been separated and increased toward the energy continuum of the nanowire. It has been shown that the lower limit of energy continuum of ±m states are split which results in different binding energy for states of +|m| and −|m| angular momentum. Finally, the oscillator strength for intersubband transitions between the system's confined energy states has been explored in order to discuss the optical properties of the described system.

Canonical quantization of dissipative systems

A consistent quantization procedure for a dissipative system is introduced. To this end Hamilton's least action principle compatible with initial value problems is exploited to obtain the Hamiltonian for the dissipative systems. The derivation of the Schrodinger equation for the dissipative system is presented. The canonical quantization method introduced is simple as well as general for linear damping. The canonical quantization procedure is employed to quantize the damped harmonic oscillator.

Exact solutions of the two-dimensional nonlinear Schrodinger equation

The nonlinear Schrödinger equation as a classical model in physics is gradually becoming an urgent problem that needs to be studied. Obtaining an exact solution of the corresponding model can not only provide theoretical support for the experiment but also provide a basis for solving practical problems. In this work, we study the propagation of waves of the two-dimensional nonlinear Schrödinger equa- tion in nonlinear media with dispersion processes. The tan-cot function method is applied. This method is effective in solving nonlinear equations of mathematical physics. Various solutions in the form of periodic waves are obtained. Graphs of the solutions arepresented.

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

AbstractEnergetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.

Considering a superposition of classical reference frames

A ubiquitous feature of quantum mechanical theories is the existence of states of superposition. This is expected to be no different for a quantum gravity theory. Guided by this consideration and others we consider a framework in which classical reference frames may be in superposition relative to one another. Mirroring standard quantum mechanics we introduce a complex-valued wavefunctional, which takes as input the transformations between the coordinates, Ψ[x(x′)], with the interpretation that an interaction between the reference frames may select a particular transformation with probability distribution given by the Born rule — P[x(x′)] = probability distribution functional ≡ |Ψ[x(x′)]|2. The cases of two and three reference frames in superposition are considered explicitly. It is shown that the set of transformations is closed. A rule for transforming wavefunctions from one system to another system in superposition is proposed and consistency with the Schrodinger equation is demonstrated.

All Charm Tetraquark Spectra in Coulombic Plus Quadratic Potential

A non-relativistic model with relativistic corrections is used to generate the mass spectra of all charm tetraquark in the diquark-antidiquark system. Fitting parameters are derived by numerically solving the Schrodinger equation for the charmonium meson using the coulombic potential and the harmonic confinement interaction potential. The mass spectra of all charm tetraquark is calculated in present work by systematically reducing a four-body problem to a two-body problem using the parameters obtained from charmonium spectra.

Spatiotemporal mode-locking of an all-fiber laser based on InP quantum dot saturable absorber

Passive mode-locking based on saturable absorbers (SAs) is an effective way to generate ultrafast pulses, while the spatiotemporal mode-locking (STML) based on SAs are rarely studied. We construct an all-fiber laser with InP quantum dots (QDs) SA, and realize multi-mode Q-switching (MMQS) and STML. In addition, by adjusting the polarization controller (PC) in the laser cavity, we obtain two different single-pulse STML states. The narrowest pulse widths of the two states are 57 ps and 32 ps, respectively, and the pulse width can be tuned continuously with increasing the pump power. By decoupling the effects of the PC and the SA in our experiment, we find that the polarization plays a key role in the selection of transverse modes. The effect of independent polarization on spatiotemporal mode-locked pulses has not been investigated before. To further analyze and understand the experimental results, we numerically solve the nonlinear Schrodinger equation, and compare the numerical simulation with the experimental results. In addition to passively adjusting cavity loss, the InP QD SA can balance the large walk off between different transverse modes, due to the spatial and spectral filtering effects. Our work has important implications for the design and application of all-fiber STML lasers.

Paul Harry Roberts. 13 September 1929—17 November 2022

Paul Roberts was a physicist and applied mathematician. He made important and often pioneering contributions in diverse research areas, but mainly with a fluid dynamic theme, which include rotating fluids, geophysical and astrophysical flows and superfluids. The cornerstone was magnetohydrodynamics with particular application to the geodynamo. His most notable achievement was the first three-dimensional self-consistent numerical geodynamo model, which built on the equations he had derived previously governing the Earth's core dynamics. His considerations included the role of thermal and compositional convection, and the inner core boundary layer (a mixed phase region). In addition to his remarkable illustration of magnetic field reversals, he applied his model to core–mantle coupling and variations in the length of the day. Paul investigated superfluid liquid helium on two fronts. He had a lifelong interest in Bose–Einstein condensates on the microscale governed by the nonlinear Schrodinger equation. His early work on the robust derivation of the mean-field Hall–Vinen–Bekharevic–Khalatnikov equations was far ahead of its time. Their importance and relevance was not appreciated until much later.

Boosting Photovoltaic Efficiency in Double Quantum Well Intermediate Band-Solar Cells through Impurity Positioning

The photovoltaic conversion efficiency of a single-intermediate band solar cell that incorporates a double quantum well structure consisting of GaAs/InAs/GaAs/InAs/GaAs embedded in the intrinsic region of conventional p-i-n structure is analyzed. The width of the intermediate band and the solutions for the two lowest energy states has been determined by solving the two-impurities-related Schrodinger equation based on the Numerov method. The position of these impurities determines three distinct cases: the system in the absence of impurities (Case 1), impurities at the center of GaAs quantum barriers (Case 2), and impurities at the center of InAs quantum wells (Case 3). The photovoltaic conversion efficiency has been calculated as a function of the widths L y H of the quantum well structures. The obtained results indicate an improvement in efficiency under the specific conditions of these parameters.

Applications of complete discrimination system approach to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation with optical soliton and bifurcation analysis

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

Bright and Dark Soliton Pulse in Solid Core Photonic Crystal Fibers

The nonlinear Schrodinger (NLS) equation was utilized to conduct an analytical study on the soliton control in homogeneous photonic crystal fibers (PCFs) within both the normal and anomalous dispersion regimes. The split step Fourier method and MATLAB computation were used to generate the analytical soliton solutions for the NLS problem. The bright and dark soliton can be controlled by the group-velocity distribution (GVD). It is capable of demonstrating the fabrication of a fully coherent PCF. Moreover, dark soliton pulses have been seen in PCF in the typical dispersion domain. Here, we present the bound states of bright–dark soliton pairs that are mutually confined in a PCF. Solitons are produced when two modes with opposing dispersions are replanted. One laser operating in the anomalous dispersion domain produces the bright soliton, while the second laser running in the normal dispersion phase produces the dark soliton by normal dispersion cross-phase modulation with the light soliton. The results unequivocally point to a novel method of generating dark soliton pulses. Capturing both bright and dark solitons can produce light states that, interestingly, have a consistent power output and spectrally resemble a frequency comb. These results may have use in soliton states in atomic physics, ultrafast optics, frequency comb technologies, and telecommunications systems. A radically new approach to the stabilization of powerful wave packages in the negative and positive GVD regions of interacting waves is illustrated.

For description of the wave (quantum) component of all living beings: A nonlinear Schrodinger equation is offered analog of the Kardar-Parisi-Zhang nonlinear mathematical model

Everything material is material body and nonlinear electromagnetic field (NEMF1), which is the basis of the duality wave-particle. All living beings have one more NEMF (NEMF2) - a weak informational field that rules and regulates everything in the body not with its strength, but with the information it carries, which makes it essential for the health of all living beings. The article first explains why the Kardar-Parisi-Zhang nonlinear mathematical model is universally applicable to all states in dynamic nonequilibrium. At the roots of this success is the fact that all dynamic events are successfully described only with nonlinear equations and Kardar-Parisi-Zhang mathematical model is a nonlinear differential equation. The model proved to be successfully applicable to a wide range of events: growing bacterial colonies, growing crystals, aggregating colloidal clusters, the dynamic of burning regions of forest or grassland, even the dynamic of eroded by acid rain statues, etc. Since everything material is a material body and a wave component - nonlinear electromagnetic field (NEMF1) and every living being is NEMF1 + NEMF2, by analogy with the Kardar-Parisi-Zhang nonlinear mathematical model, this article offers a nonlinear Schrödinger equation to describe the wave component (NEMF1) of everything material and NEMF1 + NEMF2 of every living being.

On coherent states description of exciton systems: NLSE equations and hydrodynamics

The exciton system is described by means of specially constructed coherent states (CS), according to an idea of L.V. Keldysh long ago. The system is analyzed from the geometric point of view and from the relevant symmetry group contained in the displacement operator as generador of the CS system in the Klauder–Perelomov sense. Explicitly important quantities involving the quantum statistics of the system (partition function, mean density n and the Yuen limit) are calculated and briefly discussed in relation to the known case of pure bosons (e.g. photons). The Bogoliubov type transformations for the field operators of the Wannier–Mott type exciton system are presented in an exact way (in sharp contrast with the case of kel where only a rough approximation was given) showing how these transformations induce nonlinearity (cubic at first order) in the Schrodinger equation of the system. The conditions for a hydrodynamic description of the original Keldysh model, as the possibility of a BEC (Bose–Einstein Condensation), are discussed.