The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we ca...The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.展开更多
In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical expo...In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.展开更多
We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localiz...We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.展开更多
We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota b...We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.展开更多
The well-known Schrd?inger equation is reasonably derived from the well-known diffusion equation. In the present study, the imaginary time is incorporated into the diffusion equation for understanding of the collision...The well-known Schrd?inger equation is reasonably derived from the well-known diffusion equation. In the present study, the imaginary time is incorporated into the diffusion equation for understanding of the collision problem between two micro particles. It is revealed that the diffusivity corresponds to the angular momentum operator in quantum theory. The universal diffusivity expression, which is valid in an arbitrary material, will be useful for understanding of diffusion problems.展开更多
A Schrödinger-like equation for a single free quantum particle is presented. It is argued that this equation can be considered a natural relativistic extension of the Schrödinger equation for energie...A Schrödinger-like equation for a single free quantum particle is presented. It is argued that this equation can be considered a natural relativistic extension of the Schrödinger equation for energies smaller than the energy associated to the particle’s mass. Some basic properties of this equation: Galilean invariance, probability density, and relation to the Klein-Gordon equation are discussed. The scholastic value of the proposed Grave de Peralta equation is illustrated by finding precise quasi-relativistic solutions for the infinite rectangular well and the quantum rotor problems. Consequences of the non-linearity of the proposed equation for the quantum superposition principle are discussed.展开更多
We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method...We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schr?dinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.展开更多
In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the ele...In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the electromagnetic field energy to mass and vice versa. We show that the dissipation (“conductivity”) factor and the particle implicit proper frequency are both related to the potential energy. We have also derived a new expression for the Schrödinger’s Equation considering the potential energy into this equation not as an ad hoc term, but also as an operator (Hermitian), which has the scalar potential energy as a natural eigenvalue of this operator.展开更多
In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we org...In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we organize a strong relationship among the Lie symmetries related to Klein-Gordon as well as Schr?dinger equations. Finally, we utilize the consequences of Lie point symmetries of Poisson and heat equations within Riemannian space to conclude the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation within universal Riemannian space.展开更多
Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger...Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(h^(k+1))in the H^1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^(k+1)+ τ~2) in the H^1 norm can be obtained in the Crank-Nicolson fully discrete scheme.展开更多
In this paper, we derive a new method for a nonlinear Schr ¨odinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectra...In this paper, we derive a new method for a nonlinear Schr ¨odinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in · 2norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.展开更多
The three-coupling modified nonlinear Schr?dinger(MNLS) equation with variable-coefficients is used to describe the dynamics of soliton in alpha helical protein. This MNLS equation with variable-coefficients is firstl...The three-coupling modified nonlinear Schr?dinger(MNLS) equation with variable-coefficients is used to describe the dynamics of soliton in alpha helical protein. This MNLS equation with variable-coefficients is firstly transformed to the MNLS equation with constant-coefficients by similarity transformation. And then the one-soliton and two-soliton solutions of the variable-coefficient-MNLS equation are obtained by solving the constant-coefficient-MNLS equation with Hirota method. The effects of different parameter conditions on the soliton solutions are discussed in detail. The interaction between two solitons is also discussed. Our results are helpful to understand the soliton dynamics in alpha helical protein.展开更多
A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transfor...A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transformation(DT)which is derived by Taylor expansion and determinants.We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions,singular breather and periodic wave interaction solution,singular breather and traveling wave interaction solution,bimodal breather and periodic wave interaction solution by two spectral parameters.We found a general formula for these solutions in the form of determinants.We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the(2+1)-dimensional reverse space–time nonlocal NLS equation.展开更多
Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studi...Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studied in detail.The results show that the interaction among four solitons mainly depends on the values of solution parameters;k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons;the pulse velocity and width mainly depend on the imaginary part of ki(i=1,2,3,4),while the pulse amplitude mainly depends on the real part of ki(i=1,2,3,4).展开更多
We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obt...We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.展开更多
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 exten...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, <em>i</em>.<em>e</em>. 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.展开更多
The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinet...The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.展开更多
For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each...For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.展开更多
In this article, we investigate some exact wave solutions to the higher dimensional time-fractional Schrodinger equation, an important equation in quantum mechanics. The fractional Schrodinger equation further precise...In this article, we investigate some exact wave solutions to the higher dimensional time-fractional Schrodinger equation, an important equation in quantum mechanics. The fractional Schrodinger equation further precisely describes the quantum state of a physical system changes in time. In order to determine the solutions a suitable transformation is considered to transmute the equations into a simpler ordinary differential equation (ODE) namely fractional complex transformation. We then use the modified simple equation (MSE) method to obtain new and further general exact wave solutions. The MSE method is more powerful and can be used in other works to establish completely new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics. The affect of obtaining parameters for its definite values which are examined from the solutions of two dimensional and three dimensional time-fractional Schrodinger equations are discussed and therefore might be useful in different physical applications where the equations arise in this article.展开更多
In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range pot...In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger equations.In particular,we prove the global existence when the Coulomb potential is attractive,i.e.,when K>0,and the scattering theory when the Coulomb potential is repulsive,i.e.,when K≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.展开更多
文摘The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.
基金supported by the NNSF of China(12171014, 12271539, 12171326)the Beijing Municipal Commission of Education (KZ202010028048)the Research Foundation for Advanced Talents of Beijing Technology and Business University (19008022326)。
文摘In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.
基金the National Natural Science Foundation of China(Grant Nos.11871232 and 12201578)Natural Science Foundation of Henan Province,China(Grant Nos.222300420377 and 212300410417)。
文摘We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.
基金supported by the National Natural Science Foundation of China (Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau (Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.
文摘The well-known Schrd?inger equation is reasonably derived from the well-known diffusion equation. In the present study, the imaginary time is incorporated into the diffusion equation for understanding of the collision problem between two micro particles. It is revealed that the diffusivity corresponds to the angular momentum operator in quantum theory. The universal diffusivity expression, which is valid in an arbitrary material, will be useful for understanding of diffusion problems.
文摘A Schrödinger-like equation for a single free quantum particle is presented. It is argued that this equation can be considered a natural relativistic extension of the Schrödinger equation for energies smaller than the energy associated to the particle’s mass. Some basic properties of this equation: Galilean invariance, probability density, and relation to the Klein-Gordon equation are discussed. The scholastic value of the proposed Grave de Peralta equation is illustrated by finding precise quasi-relativistic solutions for the infinite rectangular well and the quantum rotor problems. Consequences of the non-linearity of the proposed equation for the quantum superposition principle are discussed.
基金supported in part by TUT Programs on Advanced Simulation Engineering,Toyohashi University of Technology.
文摘We propose a solution method of Time Dependent Schr?dinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schr?dinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.
文摘In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the electromagnetic field energy to mass and vice versa. We show that the dissipation (“conductivity”) factor and the particle implicit proper frequency are both related to the potential energy. We have also derived a new expression for the Schrödinger’s Equation considering the potential energy into this equation not as an ad hoc term, but also as an operator (Hermitian), which has the scalar potential energy as a natural eigenvalue of this operator.
基金supported by the National Natural Science Foundation of China(Grant No.11371361)the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology(2014)the Key Discipline Construction by China University of Mining and Technology(Grant No.XZD 201602).
文摘In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we organize a strong relationship among the Lie symmetries related to Klein-Gordon as well as Schr?dinger equations. Finally, we utilize the consequences of Lie point symmetries of Poisson and heat equations within Riemannian space to conclude the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation within universal Riemannian space.
基金Project supported by the National Natural Science Foundation of China(No.11671157)
文摘Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(h^(k+1))in the H^1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^(k+1)+ τ~2) in the H^1 norm can be obtained in the Crank-Nicolson fully discrete scheme.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11271195,41231173,and 11201169)the Postdoctoral Foundation of Jiangsu Province of China(Grant No.1301030B)+1 种基金the Open Fund Project of Jiangsu Key Laboratory for NSLSCS(Grant No.201301)the Fund Project for Highly Educated Talents of Nanjing Forestry University(Grant No.GXL201320)
文摘In this paper, we derive a new method for a nonlinear Schr ¨odinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in · 2norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11874324 and 11705164)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.LY17A040011,LY17F050011,and LR20A050001)+1 种基金the Foundation of “New Century 151 Talent Engineering” of Zhejiang Province of Chinathe Youth Talent Program of Zhejiang A&F University
文摘The three-coupling modified nonlinear Schr?dinger(MNLS) equation with variable-coefficients is used to describe the dynamics of soliton in alpha helical protein. This MNLS equation with variable-coefficients is firstly transformed to the MNLS equation with constant-coefficients by similarity transformation. And then the one-soliton and two-soliton solutions of the variable-coefficient-MNLS equation are obtained by solving the constant-coefficient-MNLS equation with Hirota method. The effects of different parameter conditions on the soliton solutions are discussed in detail. The interaction between two solitons is also discussed. Our results are helpful to understand the soliton dynamics in alpha helical protein.
文摘A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transformation(DT)which is derived by Taylor expansion and determinants.We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions,singular breather and periodic wave interaction solution,singular breather and traveling wave interaction solution,bimodal breather and periodic wave interaction solution by two spectral parameters.We found a general formula for these solutions in the form of determinants.We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the(2+1)-dimensional reverse space–time nonlocal NLS equation.
基金National Natural Science Foundation of China(Grant No.11705108).
文摘Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studied in detail.The results show that the interaction among four solitons mainly depends on the values of solution parameters;k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons;the pulse velocity and width mainly depend on the imaginary part of ki(i=1,2,3,4),while the pulse amplitude mainly depends on the real part of ki(i=1,2,3,4).
基金Project supported by the National Natural Science Foundation of China(Grant No.11771151)the Guangdong Natural Science Foundation of China(Grant No.2017A030313008)+1 种基金the Guangzhou Science and Technology Program of China(Grant No.201904010362)the Fundamental Research Funds for the Central Universities of China(Grant No.2019MS110)
文摘We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.
文摘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, <em>i</em>.<em>e</em>. 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.
基金The authors gratefully acknowledge Qassim University,represented by the Deanship of Scienti c Research,on the material support for this research under the number(1671-ALRASSCAC-2016-1-12-S)during the academic year 1437 AH/2016 AD.
文摘The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.
基金supported by the Natural science foundation of China(NSF),under grand number 11071159.
文摘For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.
文摘In this article, we investigate some exact wave solutions to the higher dimensional time-fractional Schrodinger equation, an important equation in quantum mechanics. The fractional Schrodinger equation further precisely describes the quantum state of a physical system changes in time. In order to determine the solutions a suitable transformation is considered to transmute the equations into a simpler ordinary differential equation (ODE) namely fractional complex transformation. We then use the modified simple equation (MSE) method to obtain new and further general exact wave solutions. The MSE method is more powerful and can be used in other works to establish completely new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics. The affect of obtaining parameters for its definite values which are examined from the solutions of two dimensional and three dimensional time-fractional Schrodinger equations are discussed and therefore might be useful in different physical applications where the equations arise in this article.
基金The authors were supported by NSFC(12126409,12026407,11831004)the J.Zheng was also supported by Beijing Natural Science Foundation(1222019)。
文摘In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger equations.In particular,we prove the global existence when the Coulomb potential is attractive,i.e.,when K>0,and the scattering theory when the Coulomb potential is repulsive,i.e.,when K≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.