Schrodinger Equation Research Articles

Sparse adaptive basis set methods for solution of the time dependent Schrodinger equation

Scalable numerical solutions to the time dependent Schrodinger equation remain an outstanding goal in theoretical chemistry. Here we present a method which utilises recent breakthroughs in signal processing to consistently adapt a dictionary of basis functions to the dynamics of the system. We show that for two low-dimensional model problems the size of the basis set does not grow quickly with time and appears only weakly dependent on dimensionality. The generality of this finding remains to be seen. The method primarily uses energies and gradients of the potential, opening the possibility for its use in on-the-fly ab initio quantum wavepacket dynamics.

Stability, convergence and error analysis of B-spline collocation with Crank–Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

In the present study, the complex-valued Schrodinger equation (CVSE) is solved numerically by a nonic B-spline finite element method (FEM) in quantum mechanics. The approach employed is based on the collocation of nonic B-splines over spatial finite elements, so that we have continuity of the dependent variable and its first eight derivatives throughout the solution range. For time discretization, the Crank-Nicolson scheme of second order based on FEM is employed. The method is shown to be unconditionally stable and accurate to order. Three problems are considered to validate the algorithm. Comparisons are made with existing methods and analytical solutions. The proposed method exhibits good conservation properties and performs well with regards to analytical solutions for different error norms and conservative constants related to parameters in quantum classes in Physics. The computational complexity of (2N+18) arithmetic operations with the help of a nonic-diagonal matrix is also tackled by the present scheme.

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

On the Unique Continuation Property for a Coupled System of Third-order Nonlinear Schrodinger Equations

In this paper, we study the unique continuation properties for a coupled system of third-order nonlinear Schrodinger equations and show the Carleman estimates of L2 and Lp (p > 2) types, as well as exponential decay properties of the solutions. As a consequence we obtain that if (\(\mathit{u}\), \(\mathit{w}\)) = (\(\mathit{u}\)(\(\mathit{x}\), \(\mathit{t}\)), \(\mathit{w}\)(\(\mathit{x}\), \(\mathit{t}\))) is a suffciently smooth solution of the system such that there exists \(\mathit{l}\) \(\in\) \(\mathbb{R}\) with supp \(\mathit{u}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))) and supp \(\mathit{w}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))), for \(\mathit{j}\) = 1,2 (t1 \(\neq\) t2), then \(\mathit{u}\) \(\equiv\) 0 and \(\mathit{w}\) \(\equiv\) 0.

Hamiltonian Spectral Flows, the Maslov Index, and the Stability of Standing Waves in the Nonlinear Schrodinger Equation

Hamiltonian Spectral Flows, the Maslov Index, and the Stability of Standing Waves in the Nonlinear Schrodinger Equation

Highly dispersive optical solitons with cubic law and cubic-quintic-septic law nonlinearities by two methods

In this paper, the Csch technique, Sine-Cosine strategy are utilized to recover the new particular optical solitons for exceptionally dispersive dispersive Nonlinear Schrodinger Equation NLSE. These plans really get the specific arrangement of the model as dim, particular, solitary occasional and splendid particular combo optical solitons. These arrangements help the mathematicians to understand the actual peculiarities of present model. The arrangements in presents work demonstrate that these plans offer us a powerful numerical device for addressing NLSEs in different areas of applied sciences. These plans actually procure the specific arrangement of the model as dim, solitary, particular intermittent and splendid particular combo optical solitons. The arrangements in presents work demonstrate that these plans offer us a persuasive numerical device for tackling NLSEs in different areas of applied sciences.

Nanoscale Temperature Dependent Quantum-Effect Analytical Model of Short- Channel, Junction-Less, Double-Gate Stack (SC-JL-DG) MOSFET for Analog Applications at Higher Frequencies

In this paper, an in-depth temperature study is performed on Junction-less (JL), Double-Gate, metal-oxide-semiconductor field-effect transistors (JL-DGMOSFET) with gate-oxide-stack so as to deeply analyse the heating effects on the device. Quantum-effects are also induced. The effects are studied on different parameters like surface potential, threshold voltage, gm etc. An analytical model was applied to improve the efficacy of the outcomes. By adding the Schrodinger equation to the standard 2-D Poisson's equation, the analytical model is created. The conclusive expressions for Vth (change in threshold voltage) and potential are derived by employing the boundary conditions for the device. The device's parameters line up with the outcomes of analysis and simulation. The ATLAS device simulation was instrumental in the simulation work.

High-order harmonic generation, attosecond pulse, and non-sequential double ionization in the helium atom under high-intensity femtosecond laser pulses

High-order harmonic generation (HHG), attosecond pulse train (APT), isolated attosecond pulse (IAP), and non-sequential double ionization (NSDI) in the Helium atom under intense femtosecond laser pulses are computed using the time-dependent Schrodinger equation (TDSE) in one dimension (1D). By taking into account the electron–electron and electron-nucleus interactions, as well as determining the Helium atom’s ground state wave function through the imaginary time propagation (ITP) method, we are able to observe the widely recognized ‘knee structure’ in the ionization probability of the Helium atom as a function of intensity within an ionization boundary condition model. The results are in good agreement with the experimental data reported by (Walker et al (1994) Phys. Rev. Lett. 73, 1227).

Ro-vibrational energy spectra and thermal properties of some diatomic molecules

The present work deals with the solutions to the radial Schrodinger equation for harmonic plus screened Kratzer potential (HSKP) within the framework of the Nikiforov–Uvarov functional analysis method. The Greene–Aldrich approximation is used to handle the inverse square terms of the HSKP. Reducing the HSKP into Kratzer and screened Kratzer potentials, the energy spectra of selected diatomic molecules, i.e. LiH, HCl, O 2 and I 2 are computed. Further, the equations for expectation values of several parameters, including inverse of position ( r − 1 ), square of inverse of position ( r − 2 ), kinetic energy (T) and the square of momentum ( p 2 ) are obtained invoking the Hellmann–Feynman theorem. Results of this study are consistent with other similar previous studies. The analytical expressions for partition function and other thermodynamic properties of the diatomic molecules are determined.

Exact Solution of the Position-Dependent Mass Schrödinger Equation with the Completely Positive Oscillator-Shaped Quantum Well Potential

"Two exactly-solvable confined models of the completely positive oscillator-shaped quantum well are proposed. Exact solutions of the position-dependent mass Schrodinger equation corresponding to the proposed quantum well potentials are ¨ presented. It is shown that the discrete energy spectrum expressions of both models depend on certain positive confinement parameters. The spectrum exhibits positive equidistant behavior for the model confined only with one infinitely high wall and nonequidistant behavior for the model confined with the infinitely high wall from both sides. Wavefunctions of the stationary states of the models under construction are expressed through the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials appearing in wavefunctions depend on parameters a and b, but the Laguerre polynomials depend only on the parameter a. Some limits and special cases of the constructed models are discussed."

The use of Laplace transform in the inverse problem from bound states

"In a previous work dedicated to non-relativistic quantum mechanics in the three-dimensional space with spherical symmetry, we have shown the possibility of reconstructing the ground state density from its moments in the case of bound states. The method is based on relationships connecting these moments to the lower energy of each state of angular momentum `. The ground state density yields the ground state wave function, which allows us to calculate the equivalent local potential by inverting the Schrodinger equation. This calculus uses the Fourier transform of the ground state density ρ(r). The aim of this work is to check how to improve the results by replacing the Fourier transform by the Laplace transform of the quantity r2 ρ(r). The method is first presented. Then examples and results are compared to those obtained previously. "

Design of a large-mode-area tapered fiber amplifier for amplification of ultrashort laser pulses at 1535 nm wavelength

We design a W-type large-mode-area erbium doped tapered fiber with effective mode area varying from 1770 µm2 to 840 µm2with tolerance ± 0.0005 in refractive indices of different layers of fiber. The beam quality of the pulses is ensured by effective single-mode operation. We investigate the amplification of high power femtosecond pulses in this fiber by using nonlinear Schrodinger equation including gain term. It is solved using split-step Fourier method for both up-tapered (effective mode area increasing along the length) and down tapered (effective mode area decreasing along the length) fiber configurations at centered around 1535 nm wavelength. In both configurations of fiber, input pulse parameters like peak power, pulse width, and fiber length are calculated for maximum amplification with the least amount of pulse shape distortion. In down-tapered configuration, we numerically demonstrate the amplification of 1 kW peak power, 560 fs duration pulse to 3.6 MW peak power, 12 fs amplified pulse. In up-tapered configuration, we obtain the amplification of 2 kW, 565 fs input pulse to ∼7.9 MW peak power, ∼11 fs pulse. The fiber is useful for amplifying high-power laser pulses, and the resulting pulses can be used for laser processing, spectroscopy, and remote sensing.

Comparative Study of the Mass Spectra of Heavy Quarkonium System with an Interacting Potential Model

In this work, the comparison of the mass spectra of heavy quarkonium system with an interacting potential (Class of Yukawa potential) was studied. The Schrodinger equation is analytically solved using Nikiforov- Uvarov (NU) method and series expansion method (SEM). The approximate solutions of the eigen energy equation and corresponding eigenfunction in terms of Laguerre polynomials were obtained using the NU method and the solutions of the eigen energy equation were also obtained with the SEM. The mass spectra for heavy quarkonium system (HQS) for the potential under study were obtained for bottomonium and charmonium HQS. We compared the results obtained between NU and SEM. It was noticed that SEM solutions yield mass spectra very close to experimental data compared to solutions with NU method. The obtained results were also compared with works by some other authors and were found to be improved. This study can be extended by using other exponential-type potential models with other analytical approach and a different approximation schemes to obtain the mass spectra of heavy quarkonium system. The relativistic properties using Klein-Gordon or Dirac equations can be explored to obtain the mass spectra of light quarkonia. Finally, the information entailed in the normalized wave-functions can also be studied.

ON THE PROBABILITY OF UNIDIRECTIONAL NONLINEAR EXTREME WAVES IN THE PRESENCE OF WAVE REFLECTION

Extreme waves are known to appear in different water depth regimes, and their focusing mechanisms have been intensively studied over the last decades. This experimental study aims to improve understanding of rogue wave statistics when wave reflections are at play. A series of unidirectional JONSWAP wave trains have been generated in a wave flume while varying the water depth and beach inclination slope. The data collected near the beach installation suggests a general decrease in extreme wave probability with the increase of the beach inclination, thus, with the increase of wave reflection. In deep-water, the numerical simulations based on the coupled nonlinear Schrodinger equation, which accounts for the presence of reflective waves, show a very good agreement with the experimental data. The irregular wave measurements in the presence of wave reflection indicate that the decrease in probability of exceedance, which correlates with the decrease in the value of kurtosis, is due to the weakening of third-order effects. When the water depth is decreased to a finite depth regime, we can observe an increase of extreme events, which we attribute to the change of the wave breaking threshold. However, no substantial rogue waves have been observed below the dimensionless depth kh value of 1.36. Then again, nonlinear effects always remain relevant and at play in all cases considered.

Magnetic exchange field signature in ultrafast electron dynamics in monolayer WSe2 by ultrashort optical pulse

We theoretically investigate the ultrafast electron dynamics in monolayer tungsten diselenide interacting with an irradiated ultrashort strong optical pulse in the presence of a magnetic exchange field. The dynamics of Dirac fermions in monolayer WSe2 is affected by spin and valley splitting, of course, enhanced by applied magnetic exchange field. There exists remarkable interaction between the exchange field and valley index because of huge spin–orbit coupling at the valence band. Regarding the optical pulse duration is smaller than the electron scattering time, the electron dynamics remains coherent and is described by the time-dependent Schrodinger equation. The strong electric field of the pulse, interacted with monolayer WSe2 causes appearance of a dipole moment connecting two valence and conduction bands. Because of the K and K′ valleys behave differently by the magnetic exchange field, the perfect valley-spin polarization effect is achieved. We explore the irreversible conduction band electron transition affected by a magnetic exchange field. The signature of the exchange field on such transitions is quite obvious. The conduction band electron redistribution around the Dirac points is represented, when the pulse ends. Asymmetric hot spots pattern in two different K and K′ valleys depends more or less on the magnitude of the magnetic exchange field. By reversing the direction of the exchange field, the process of spin polarization is completely reversed and it results in the electron distribution pattern. The high spin-valley resolved splitting in monolayer WSe2 provides a unique platform to investigate the magnetic exchange field driven phenomena, leading to valley polarization effect.

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.

A prince for the sleeping beauty - NFT for soliton signal processing

The soliton, the eigen-solution of nonlinear Schrodinger equation (NLSE), has been investigated in many branches of physics. The first prediction of a soliton in optical fiber could be dated back to 1973. Since then, numerous theoretical and experimental works on solitons have been carried out, including soliton communication and soliton generation in fiber lasers. However, due to the constraints of hardware and digital signal processing methods, research on solitons have been stalled for decades. Recently, the nonlinear Fourier transform (NFT) has attracted significant attention in the field of optical fiber communications and soliton-based signal processing, which provides a powerful mathematical framework for analyzing and understanding the behavior of solitons in optical fibers, enabling the development of novel signal processing techniques. The NFT technique has brought the soliton back to life. It has been widely recognized that in nonlinear frequency domain, the nonlinear crosstalk resulting from the Kerr effect is practically negligible, and fiber nonlinearity damage caused by the Kerr effect can be accounted for as a linear transform factor. Moreover, NFT has been demonstrated for the analysis of laser radiation, highlighting its potential in characterizing ultrafast pulses in the nonlinear frequency domain. In this submission, we review the advancements of NFT concerning both in soliton communication and characterization, focusing on practical implementation aspects. With many practical implementation aspects still being open, our mini review is aimed to assist researchers in evaluating the potential, identifying challenges, and envisioning future directions in NFT-based signal processing technologies. And wake up the sleeping beauty – the soliton!

Accurate Computation of b-Coefficients in Nonlinear Fourier Transform for the Manakov Equation

In recent years, nonlinear Fourier transform (NFT) has been intensely researched as a scheme for communication over the nonlinear optical fiber and characterizing dynamics in nonlinear optical systems. For the accurate computation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients on discrete eigenvalues, the bidirectional algorithm has been proposed where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients are calculated at a cutting point within the temporal window rather than two boundaries. The performance of the bidirectional algorithm depends severely upon the selection criterion of the cutting-point, which has been investigated in nonlinear Schrodinger equation and Korteweg–De Vries equation but not yet in Manakov equation accounting for the dual-polarization signals over optical fiber. In this paper, we propose a cutting-point criterion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients for the bidirectional algorithm of Manakov equation and compare it with existing criteria. Numerical results show that, the resulting algorithm has much better accuracy compared with existing criteria, especially for signals containing a large number of eigenvalues with large imaginary parts. The bidirectional algorithm with the proposed criterion is useful for decoding the dual-polarization nonlinear frequency division multiplexing (DP-NFDM) signals for communication and characterization of vector soliton dynamics in Manakov equation described systems.

Asymmetric tunneling of holes through a semiconductor junction in an arbitrary magnetization configuration

In this paper, we investigate the asymmetry of holes tunneling through GaMnAs/GaAs/GaMnAs heterostructures in the cases general, where the magnetization of the two electrodes are not colinear and their magnitude are not equal. The six-bands $\textbf{k.p}$ Hamiltonian is employed to describe the holes in the GaAs and GaMnAs layers, taking into account both spin-orbit and exchange interactions. The multi-band transfer-matrix formalism is applied for numerically solving the Schrodinger equation to derive the hole wave transmission. We then calculate the transmission asymmetry and discuss its dependence on the right electrode magnetization magnitude, as well as the angles determining relatively the orientation of two magnetization vectors. The study may provide the insights into dynamic of Anomalous Tunneling Hall current during magnetization switching process which are important in spintronics technology for device design and measurement.

Study of a nonlinear Schrodinger equation with truncated M proportional derivative

In this work, we introduce a novel truncated M proportional (T-MP) derivative and consider a T-MP perturbed derivative nonlinear Schrodinger equation (PDNLSE). The PDNLSE is a nonlinear model which arises in nano optical fibers (photonic nanowires). While the DNLSE exhibits self-steepening (SS), the PDNLSE exhibits self-phase modulation (SPM) and Raman scattering (RS) effects, Here, the exact solutions of the PDENLSE are derived, here, by implementing the unified method. These solutions are displayed in graphs. It is found that a sufficient condition for a hyper-chaotic solution to hold is that an elliptic function solution exists. Non elliptic solutions may be, also, hyper-chaotic. It is worth mentioning that hyper chaotic may occur in economic and financial mathematics. Nonchaotic solutions exhibit many geometric structures, chirped solitons, and rhombus-shaped and M-shaped solitons. It is found that modulation instability triggers when the coefficient of the third-order dispersion exceeds a critical value. Further, the global bifurcation is investigated via phase portrait by constructing the Hamiltonian function.