One-dimensional Schrodinger Equation Research Articles

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Transformation operator for the Schrodinger equation with additional exponential potential

In this paper, we consider the one-dimensional Schrodinger equation on the semiaxis with an additional exponential potential. Using transformation operators with the asymptotics at infinity, a triangular representation of a special solution of this equation is found. An estimate is obtained with respect to the kernel of the representation

Numerical simulation of one-dimensional nonlinear Schrodinger equation using PSO with exponential B-spline

Particle swarm optimization (PSO) is a widely adopted, rapid, efficient, and computer-friendly technique for optimization in various fields, including computer science, engineering, and mathematical modelling. In this paper, the nonlinear Schrodinger (NLS) equation has been solved numerically, which leads to soliton-type solutions, by utilizing the “Exponential cubic B-spline differential quadrature method (Expo-MCB-DQM) with PSO”. The exponential cubic B-spline basis functions involve a parameter that must be assigned a value to find the numerical solutions. Until now, the value of a parameter had been determined by the hit-and-trial approach, which leads to unstable results. By employing the PSO technique, the optimal value of the parameter is determined, which enhances the accuracy of the numerical solutions. This study introduces an innovative combination of Expo-MCB-DQM with PSO, which is an innovative combination that optimizes the parameters in the basis function, leading to improved outcomes. To validate the authenticity and effectiveness of this methodology, it has been applied to two problems to present the numerical results in comparison with the exact solution. Additionally, the paper provides insightful discussions on the significant application of the NLS equation in optical fiber communication.

Spin-dependent high-order topological states and corner modes in a monolayer FeSe/GdClO heterostructure

We propose that a spin-dependent second-order topological insulator can be realized in monolayer FeSe/GdClO heterostructure, in which substrate GdClO helps to stabilize and enhance the antiferromagnetic order in FeSe. The second-order topological insulator is free from spin-orbit coupling and in-plane magnetic field. We also find that there exist two types of distinct corner modes residing in intersections of two ferromagnetic edges and two antiferromagnetic edges, respectively. The underlying physics for ferromagnetic corner mode follows a sublattice-chirality-kink picture. More interestingly, ferromagnetic corner mode shows spin-dependent property, which is also robust against spin-orbit coupling. Unexpectedly, antiferromagnetic corner mode can be taken as a typical emergent and hierarchical phenomenon from an array of ferromagnetic corner modes. Remarkably, antiferromagnetic corner modes violate general kink picture and can be understood as bound states of a one-dimensional Schrodinger equation under a connected potential well. Our findings not only provide a promising second-order topological insulator in electronic materials, but uncover some new properties of corner modes in high-order topological insulator.

SUSY-nonrelativistic quantum eigenspectral energy analysis for squared-type trigonometric potentials through Nikiforov-Uvarov formalism

Explicit and analytical bound-state solutions of the Schrodinger equation for squared-form trigonometric potentials within the framework of supersymmetric quantum mechanics (SUSYQM) are performed by implementing the Nikiforov-Uvarov (NU) polynomial procedure. The first step requires a certain action to adopt an appropriate ansatz superpotential W(x) for generating the potential pair as V1 (x) and V2(x). In the second process, inserting each potential for the one-dimensional Schrodinger equation and solving the hypergeometric differential equation with the NU method gives rise to normalized wave function descriptions and algebraically corresponds to the characteristic SUSY quantum energy eigenspectrum sets. It is remarkable to note that, when examined parametrically, they are of reliable and applicable forms concerning the mathematical treatment of various physical quantum systems prescribed in relativistic or nonrelativistic contexts.

Bound-state soliton gas as a limit of adiabatically growing integrable turbulence

We study numerically the integrable turbulence in the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type using a new approach called the “growing of turbulence”. In this approach, we add a small linear pumping term to the equation and start evolution from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and examine the resulting integrable turbulence. For sufficiently small initial noise and pumping coefficient, and also for not very wide simulation box (basin length), we observe that the turbulence grows in a universal adiabatic regime, moving successively through the statistically stationary states of the integrable 1D-NLSE, which do not depend on the pumping coefficient, amplitude of the initial noise or basing length. Waiting longer in the growth stage, we transit from weakly nonlinear states to strongly nonlinear ones, characterized by a high frequency of rogue waves. Using the inverse scattering transform (IST) method to monitor the evolution, we observe that the solitonic part of the wavefield becomes dominant even when the (linear) dispersion effects are still leading in the dynamics and with increasing average intensity the wavefield approaches a dense bound-state soliton gas, whose properties are defined by the Fourier spectrum of the initial noise. Regimes deviating from the universal adiabatic growth also lead to solitonic states, but solitons in these states have noticeably different velocities and a significantly wider distribution by amplitude, while the statistics of wavefield indicates a much more frequent appearance of very large waves.

Low temperature photoluminescence study for identification of intersubband energy levels inside triangular quantum well of AlGaN/GaN heterostructure

A non-destructive method for extracting the true signatures of intersubband energy level states inside a single triangular quantum well for AlGaN/GaN heterostructure is presented. Herein, we report experiments showing light interaction with high-mobility two-dimensional electron gas (2DEG) in Al0.3Ga0.7N/AlN/GaN based HEMT structure using low-temperature photoluminescence spectroscopy. An interband transition from two-dimensional electron gas (2DEG) subbands to the valence band is identified. These observed transitions were confirmed by comparing the PL spectra of the as grown and top barrier layer etched samples. The luminance peak related to 2DEG disappeared when the top AlGaN barrier layer was removed using reactive ion etching (RIE) system. Furthermore, the emission peak data are also supported by a calculation based on a self-consistent solution of one-dimensional Poisson and Schrodinger equations. The three PL spectra peaks corresponding to the interband transitions from 2DEG subbands to the valence band were reported at 3.269, 3.356 and 3.438 eV respectively. The corresponding intersubband energy states inside the quantum well were extracted (simulated) with 87 (91) and 178 (186) meV energy spacing between E0 & E1 and E0 & E2 respectively. The temperature and excitation power-dependent PL measurement makes it easy to identify the transitions from the 2DEG subbands to valence bands.

Approximate Solutions of the Schrodinger Equation for a Momentum-Dependent potential

The solution of one-dimensional Schrodinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally, the special cases of the interacting potential were obtained.

Ferromagnetic Resonance Modes in the Exchange-Dominated Limit in Cylinders of Finite Length

We analyze the magnetic mode structure of axially magnetized finite-length nanoscopic cylinders in a regime where the exchange interaction dominates, along with simulations of the mode frequencies of the ferrimagnet yttrium iron garnet. For the bulk modes, we find that the frequencies can be represented by an expression given by Herring and Kittel by using wavevector components obtained by fitting the mode patterns emerging from these simulations. In addition to the axial, radial, and azimuthal modes that are present in an infinite cylinder, we find localized ``cap modes'' that are ``trapped'' at the top and bottom cylinder faces by the inhomogeneous dipole field emerging from the ends. Semiquantitative explanations are given for some of the modes, in terms of a one-dimensional Schrodinger equation, which is valid in the exchange-dominant case. The assignment of the azimuthal-mode number is carefully discussed, and the frequency splitting of a few pairs of nearly degenerate modes is determined through the beat pattern emerging from them.

On the Analytical and Numerical Solutions of the One-Dimensional Nonlinear Schrodinger Equation

In this paper, four compelling numerical approaches, namely, the split-step Fourier transform (SSFT), Fourier pseudospectral method (FPSM), Crank-Nicolson method (CNM), and Hopscotch method (HSM), are exhaustively presented for solving the 1D nonlinear Schrodinger equation (NLSE). The significance of this equation is referred to its notable contribution in modeling wave propagation in a plethora of crucial real-life applications such as the fiber optics field. Although exact solutions can be obtained to solve this equation, these solutions are extremely insufficient because of their limitations to only a unique structure under some limited initial conditions. Therefore, seeking high-performance numerical techniques to manipulate this well-known equation is our fundamental purpose in this study. In this regard, extensive comparisons of the proposed numerical approaches, against the exact solution, are conducted to investigate the benefits of each of them along with their drawbacks, targeting a broad range of temporal and spatial values. Based on the obtained numerical simulations via MATLAB, we extrapolated that the SSFT invariably exhibits the topmost robust potentiality for solving this equation. However, the other suggested schemes are substantiated to be consistently accurate, but they might generate higher errors or even consume more processing time under certain conditions.

Solitonic model of the condensate

We consider a spatially extended box-shaped wave field that consists of a plane wave (the condensate) in the middle and equals zero at the edges, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. Within the inverse scattering transform theory, the scattering data for this wave field is presented by the continuous spectrum of the nonlinear radiation and the soliton eigenvalues together with their norming constants; the number of solitons N is proportional to the box width. We remove the continuous spectrum from the scattering data and find analytically the specific corrections to the soliton norming constants that arise due to the removal procedure. The corrected soliton parameters correspond to symmetric in space N-soliton solution, as we demonstrate analytically in the paper. Generating this solution numerically for N up to 1024, we observe that, at large N, it converges asymptotically to the condensate, representing its solitonic model. Our methods can be generalized for other strongly nonlinear wave fields, as we demonstrate for the hyperbolic secant potential, building its solitonic model as well.

On Hydrogen Entanglements and Gravitation: a Lie Symmetry Approach

We depart from the popular view on how gravitation is generated. Ours is entanglement based. For an atom, hydrogen in this case, all subatomic particles, within and outside the nuclei, participate in this eternal dance. To test the idea, we use it to determine a formula for G, the universal gravitational constant. In the process, we note that the one-dimensional Schrodinger equation is not solved, as claimed. For example, existing solutions are silent on the quantum superposition principle. This we address through our modified symmetry group theoretical methods.

Extreme rogue wave generation from narrowband partially coherent waves.

In the framework of the focusing one-dimensional nonlinear Schrödinger equation, we study numerically the integrable turbulence developing from partially coherent waves (PCW), which represent superposition of uncorrelated linear waves. The long-time evolution from these initial conditions is characterized by emergence of rogue waves with heavy-tailed (non-Gaussian) statistics, and, as was established previously, the stronger deviation from Gaussianity (i.e., the higher frequency of rogue waves) is observed for narrower initial spectrum. We investigate the fundamental limiting case of very narrow initial spectrum and find that shortly after the beginning of motion the turbulence enters a quasistationary state (QSS), which is characterized by a very slow evolution of statistics and lasts for a very long time before arrival at the asymptotic stationary state. In the beginning of the QSS, the probability density function (PDF) of intensity turns out to be nearly independent of the initial spectrum and is very well approximated by a certain Bessel function that represents an integral of the product of two exponential distributions. The PDF corresponds to the maximum possible stationary value of the fourth-order moment of amplitude κ_{4}=4 and yields a probability to meet intensity above the rogue wave threshold that is higher by 1.5 orders of magnitude than that for a random superposition of linear waves. We routinely observe rogue waves with amplitudes ten times larger than the average one, and all of the largest waves that we have studied are very well approximated by the amplitude-scaled rational breather solutions of either the first (Peregrine breather) or the second orders.

On One Implementation of the Numerov Method for the One-Dimensional Stationary Schrödinger Equation

We present accurate numerical results for the one-dimensional stationary Schrodinger equation in the case of three quantum problems: quantum harmonic oscillator, radial Schrodinger equation for a Hydrogen atom, and a particle penetration through the potential barrier. All of them were solved by the Numerov method with high accuracy and we plot their wave functions using the results of numerical calculations. Furthermore, we offer accurate numerical methods for solving boundary value problems, eigenvalue problems, matrix elimination.

Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

<p style='text-indent:20px;'>In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential <inline-formula><tex-math id="M1">\begin{document}$ u(\omega t) $\end{document}</tex-math></inline-formula>. We show that if the frequency vector <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula>. In contrast with previous results, the conditions of small potential <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> or large energy <inline-formula><tex-math id="M5">\begin{document}$ E $\end{document}</tex-math></inline-formula> are no longer needed.

Solving the Schrodinger Equation on the Basis of Finite-Difference and Monte-Carlo Approaches

The paper presents a method of numerical solution of the Schrodinger equation, which combines the finite-difference and Monte-Carlo approaches. The resulting method was effective and economical and, to a certain extent, not improved, i.e. optimal. The method itself is formalized as an algorithm for the numerical solution of the Schrodinger equation for a molecule with an arbitrary number of quantum particles. The method is presented and simultaneously illustrated by examples of solving the one-dimensional and multidimensional Schrodinger equation in such problems: linear one-dimensional oscillator, hydrogen atom, ion and hydrogen molecule, water, benzene and metallic hydrogen.

ANALYSIS OF ENERGY AND WAVE FUNCTIONS AND THE THERMODYNAMICS PROPERTIES OF THE 6-DIMENSIONAL SCHRODINGER EQUATION UNDER DOUBLE RING-SHAPE OSCILLATOR (DRSO) AND MANNING-ROSEN POTENTIALS USING SUSY QM METHOD

The exact solutions of the Schrodinger equations (SE) in the D-dimensional coordinate system have attracted the attention of many theoretical researchers in branches of quantum physics and quantum chemistry. The energy eigenvalues and the wave function are the solutions of the Schrodinger equation that implicitly represents the behavior of a quantum mechanical system. This study aimed to obtain the eigenvalues, wave functions, and thermodynamic properties of the 6-Dimensional Schrodinger equation under Double Ring-Shaped Oscillator (DRSO) and Manning-Rosen potential. The variable separation method was applied to reduce the one 6-Dimensional Schrodinger equation depending on radial and angular non-central potential into five onedimensional Schrodinger equations: one radial and five angular Schrodinger equations. Each of these onedimensional Schrodinger equations was solved using the SUSY QM method to obtain one eigenvalue and one wave function of the radial part, five eigenvalues, and five angular wave functions angular part. Some thermodynamic properties such, the vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆, were obtained using the radial energy equations. The results showed that except the 𝑛𝑙1, all increment of angular quantum number decreases the energy values. Increments of all potential parameter increase the energy values. Increment of angular quantum number and potentials parameter increases the amplitude and shifts the wave functions to the left. However, the increment of 𝑛𝑙1, 𝛼, 𝜎, and 𝜌 decrease the amplitude and shift wavefunctions to the right. Moreover, the vibrational mean energy 𝑈 and free energy 𝐹 increased as the increasing value of potentials parameters, where the ω parameter has the dominant effect than the other parameters. The vibrational specific heat 𝐶 and entropy 𝑆 affected only by the 𝜔 parameter, where 𝐶 and 𝑆 decreased as the increase of 𝜔.

Energy Spectrum in a Shallow GaAs/AlGaAs Quantum Well Probed by Spectroscopy of Nonradiative Broadening of Exciton Resonances

The energy spectrum of the exciton and carrier states in a shallow GaAs/AlGaAs quantum well is experimentally studied by means of the spectroscopy of the nonradiative broadening of exciton resonances and the spectroscopy of the photoluminescence excitation. The observed peculiarities of the spectra are treated using the numerical solution of the one-dimensional Schrodinger equation for free carriers and three-dimensional equation for excitons in the quantum well. The conduction and valence band offsets in the shallow GaAs quantum well are determined.

Role of external fields on the nonlinear optical properties of a n-type asymmetric $$\delta$$-doped double quantum well

The effects of in-growth applied electric fields and in-plane (x-oriented) magnetic fields on the nonlinear optical rectification (NOR), second harmonic generation (SHG) and third harmonic generation (THG) of n-type asymmetric double $$\delta$$ -doped GaAs quantum well are theoretically investigated. One-dimensional Schrodinger equation is solved by considering effective mass and parabolic band approximations to obtain subband energy levels and their related wave functions. The variations in the NOR, SHG and THG coefficients are determined by using the iterative solutions of the compact density matrix approach. Obtained results indicate that the applied electric field leads to optical red-shift on NOR, SHG and THG coefficients while the magnetic field causes optical blue-shift on that coefficients. Hence we can conclude that applied electromagnetic fields can be used to tune optical properties of devices working within the region of infrared electromagnetic spectrum.

A new implicit high-order six-step singularly P-stable method for the numerical solution of Schrödinger equation

In this paper, we present a new implicit six-step singularly P-stable method with vanished phase-lag and its derivatives up to fifth order for the numerical integration of the one-dimensional radial time independent Schrodinger equation. The periodicity region of the method is plotted and the numerical stability and phase properties of the new methods are analyzed. The advantage of the new method in comparison with similar methods—in terms of efficiency, accuracy and stability—have been shown by implementing them in the radial time-independent Schrodinger equation during the resonance problems with the use of the Woods–Saxon potential.