Analytical Solutions of a Model for Brownian Motion in the Double Well Potential

In this paper, the analytical solutions of Schr¨odinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker–Planck equation known as the Klein–Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr¨odinger equation. The analytical results obtained from the two different methods agree with each other well. The double well potential is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.     

Potential energy surface and formation of superheavy nuclei with the Skyrme energy-density functional

With the Skyrme energy-density functional theory, the nucleus–nucleus potential is calculated and the potential energy surface is obtained with different effective forces for accurately estimating the formation cross sections of superheavy nuclei in massive fusion reactions. The width and height of the potential pocket are influenced by the Skyrme effective forces SkM, SkM^*, SkP, SIII, Ska, and SLy4, which correspond to the different equations of state for the isospin symmetry nuclear matter. It is found that the nucleus–nucleus potential is associated with the collision orientation and Skyrme forces. A more repulsive nuclear potential is pronounced with increasing the incompressible modulus of nuclear matter, which hinders the formation of superheavy nuclei. The available data in the fusion-evaporation reaction of ⁴⁸Ca+²³⁸U are nicely reproduced with the SkM^* parameter by implementing the potential into the dinuclear system model.     

Exact Solutions of the Morse-like Potential,Step-Up and Step-Down Operators via Laplace Transform Approach

We intend to realize the step-up and step-down operators of the potential V (x) = V₁ e ^2βx + V₂ e ^βx. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential (β → -β).     

Effects of Parametrization of A Woods—Saxon Potential on Single Particle Orbits in 66Ca

A Woods—Saxon potential is introduced to serve as initial conditions for an iterative solution of the exotic nuclei in relativistic mean field approach. Effects of parametrization of the Woods—Saxon potential on single particle orbits around the Fermi surface are investigated in ⁶⁶Ca. A semi-parabolic curve of orbital radius appears when the width of the central potential changes. The pseduospin splitting is more sensitive to the potential width than to the depth and the diffuseness. The spin-orbit splitting is more sensitive to the potential depth than to the width and the diffuseness.     

Approximate Bound States Solution of the Hellmann Potential

The Hellmann potential, which is a superposition of an attractive Coulomb potential -a/r and a Yukawa potential b e^-δr/r, is often used to compute bound-state normalizations and energy levels of neutral atoms. By using the generalized parametric Nikiforov-Uvarov (NU) method, we have obtained the approximate analytical solutions of the radial Schrödinger equation (SE) for the Hellmann potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented, which show good agreement with a numerical amplitude phase method and also those previously obtained by other methods. As a particular case, we find the energy levels of the pure Coulomb potential.     

Bound states resulting from interaction of the non-relativistic particles with the multiparameter potential

In this study, we present the analytical solutions of bound states for the Schrodinger equation with the multiparameter potential containing the different types of physical potentials via the asymptotic iteration method by applying the Pekeristype approximation to the centrifugal potential. For any n and l(states) quantum numbers, we derive the relation that gives the energy eigenvalues for the bound states numerically and the corresponding normalized eigenfunctions. We also plot some graphics in order to investigate effects of the multiparameter potential parameters on the energy eigenvalues.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

A Method to Solve the Schrödinger Equation for Any Power Hypercentral Potentials

An exact closed form of solution to the hyperradial Schrödinger equation is constructed for any general case comprising any hypercentral power and inverse-power potential. The hypercentral potential depends only on the hyperradius, which itself is a function of Jacobi relative coordinates that are functions of particle positions (r₁,r₂,…, r_N). This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ=power series× exp(polynomial)=[f(x)exp(g(x))] is potentially a solution of the Schrödinger equation, where the polynomial g(x) is an ansatz depending on the interaction potential.     

基于模糊控制的VSVPWM中点电位平衡策略

针对中点钳位型(NPC)三电平逆变器存在的直流侧中点电位不平衡问题,提出了一种基于模糊控制 的中点电位平衡策略。该策略重新定义了虚拟空间矢量,使虚拟小矢量不引起中点电位波动,并在虚拟中矢量中 引入了对中点电位不平衡有抑制作用的控制因子 Q。同时,设计了由模糊控制器 A 和 B 构成的组合模糊控制器, 该控制器能够依据中点电位偏移情况调整控制因子 Q,实现对中点电位的闭环控制。仿真和实验结果表明,采用 该策略的 NPC 型三电平逆变器中点电位不平衡问题能得到有效改善。     

The bound state solution for the Morse potential with a localized mass profile

We investigate an analytical solution for the Schr o¨dinger equation with a position-dependent mass distribution, with the Morse potential via Laplace transformations. We considered a mass function localized around the equilibrium position.The mass distribution depends on the energy spectrum of the state and the intrinsic parameters of the Morse potential. An exact bound state solution is obtained in the presence of this mass distribution.     

Solutions of the SchrödingerEquation forPT-Symmetric Coupled Quintic Potentialsin Two Dimensions

This paper deals with the solutions of time independent Schrödinger wave equation for a two-dimensional PT-symmetric coupled quintic potential in its most general form. Employing wavefunction ansatz method, general analytic expressions for eigenvalues and eigenfunctions for first four states are obtained. Solutions of a particular case are also presented.     

Analytic Results in the Position-Dependent Mass Schrödinger Problem

We investigate the Schrödinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass V(x)=0 case whose solutions are hypergeometric functions in tanh²x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form V(x)=V₀ sinh²x.     

Effect of confined one-gluon-exchange potential and instanton-induced interaction on nucleon–nucleon interaction

The effect of confined one-gluon-exchange potential and instanton-induced interaction potential in the singlet(~1S_0) and triplet(~3S_1) channels for nucleon–nucleon interaction has been investigated in the framework of the relativistic harmonic model using the resonating group method in the adiabatic limit with the Born–Oppenheimer approximation. The contributions of the different components of the interaction potentials have been analyzed.     

Bose-Einstein Condensate in a Linear Trap with a Dimple Potential

We study Bose-Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrdinger equation is used instead of the Gross-Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.     

利用电子结构数据拟合H2分子的电子壳模型势函数参数

电子壳模型势函数在离子晶体的原子级计算机模拟中有广泛应用,其势参数主要通过拟合晶体的实验数据或电子结构数据得到.提出了通过拟合双原子分子的量子化学从头计算电子结构数据来获得该势函数的方法,并由H₂分子的电子结构数据建立了H原子间的电子壳模型势函数.此外,还应用该势函数对H⁺₂分子离子进行了计算.该势函数拟合方案更适合于共价键型的分子. 关键词： 电子壳模型势 参数拟合 共价键 2分子')" href="#">H₂分子     

Proton radioactivity within the generalized liquid drop model with various versions of proximity potentials

In the present work, we systematically investigate the proton radioactivity half-lives of spherical proton emitters adopting a generalized liquid drop model(GLDM) with 16 different proximity potentials, of which the proximity potential Prox.77-13 gives the closest results to the experimental data. Combined with the previous conclusion that the GLDM with proximity potential Prox.77-13 can also best describe α decay half-lives, which makes the model more uniform and consistent.Further, we use the proximity potential Prox.77-13 in GLDM to predict the proton radioactivity half-lives of 14 spherical proton emitters that are allowed energetically but not yet observed experimentally or specifically quantified. Finally, we research the Geiger-Nuttall law for proton radioactivity. The results reveal that the Geiger-Nuttall law can also be well used to study the proton radioactivity half-lives of isotopes with the same orbital angular momentum l.     

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

Parameterization of Nuclear Hulthén Potential for Nucleus-Nucleus Elastic Scattering

In contrast to Gaussian or Woods-Saxon potential a two-term four parameter nuclear Hulth′en type interaction is considered to describe the α-α, α-~3He and α-~3H systems. By exploiting the phase function method, scattering phase shifts are computed up to ELab = 100 MeV for the α-α system and ELab = 15 Me V for α-~3He and α-~3H systems.The S-wave phase shift δ_0 for the α-α system tends to 2π and δ_(3/2)-for the α-~3He system tends to π, in the limit of zero energy. Reasonable agreements in phase shifts with the standard data are obtained with this simple potential model except for the 5/2~- states of α-~3He and α-~3H systems. With an additional energy-dependent correction factor to our potential, a good agreement with experimental data is obtained for 5/2~- states. We have also compared our results with the convenient Born approximations.     

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form, V(r,θ)=V(r)+(?²/2Mr²) [(βsin²θ +γcos²θ +λ)/sin θcosθ]² and its exact solutions are presented via the Nikiforov-Uvarov method. The angle dependent part V(θ)=(?²/2Mr²) [(βsin²θ +γcos²θ +λ)/sin θcosθ]², which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.     

Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential

In this work, we determine the Fisher and Shannon entropies, the expectation values and the squeeze state for a noncentral inversely quadratic plus exponential Mie-type potential analytically.The proposed potential is solved under the Schr?dinger equation using a special Greene Aldrich approximation to the centrifugal term to obtain a normalised wave function within the framework of the Nikiforov–Uvarov method. Numerical results are obtained for different screening parameters:α?=?0.1, 0.12 and 0.13 for varying real constant parameter(B). The numerical solutions are obtained only for ground state. The numerical results of Fisher entropy both for position and momentum spaces are in good agreement with existing literature. The normalisation constant, wave function, and probability density plots are carried out using a well designed Mathematica algorithm.The Fourier transform of position space entropy gives the momentum space entropy.