An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective ha...An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective hamiltonian deviates form the actual one by the presence of acorrelation operator.The cprrelation operatop is given in an explicit form.展开更多
By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can b...By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can benaturally obtained.The characteristic polynomial theory has been fully employed in our derivation.展开更多
In this paper,we formulate and analyse a kind of parareal-RKN algo-rithms with energy conservation for Hamiltonian systems.The proposed algorithms are constructed by using the ideas of parareal methods,Runge-Kutta-Nys...In this paper,we formulate and analyse a kind of parareal-RKN algo-rithms with energy conservation for Hamiltonian systems.The proposed algorithms are constructed by using the ideas of parareal methods,Runge-Kutta-Nystr¨om(RKN)methods and projection methods.It is shown that the algorithms can exactly pre-serve the energy of Hamiltonian systems.Moreover,the convergence of the integra-tors is rigorously analysed.Three numerical experiments are carried out to support the theoretical results presented in this paper and show the numerical behaviour of the derived algorithms.展开更多
Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are establi...Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the ...The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the continuous finite element method （CFEM） belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.展开更多
Based on the principle of total energy conservation, we give two important algorithms, the total energy conservation algorithm and the symplectic algorithm, which are established for the spherical shallow water equati...Based on the principle of total energy conservation, we give two important algorithms, the total energy conservation algorithm and the symplectic algorithm, which are established for the spherical shallow water equations. Also, the relation between the two algorithms is analyzed and numerical tests show the efficiency of the algorithms.展开更多
A class of quasi-exact solutions of the Rabi Hamiltonian,which describes a two-level atom interacting witha single-mode radiation field via a dipole interaction without the rotating-wave approximation,are obtained by ...A class of quasi-exact solutions of the Rabi Hamiltonian,which describes a two-level atom interacting witha single-mode radiation field via a dipole interaction without the rotating-wave approximation,are obtained by using awavefunction ansatz.Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and thefield frequency satisfy certain relations.As an example,the lowest exact energy level and the corresponding atom-fieldentanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes-Cummings andcounter-rotating cases of the Rabi Hamiltonian.展开更多
By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved hav...By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.展开更多
Theoretical solid-state physicists formulate their models usually in the form of a Hamiltonian. In quantum mechanics, the Hamilton operator (Hamiltonian) is of fundamental importance in most formulations of quantum th...Theoretical solid-state physicists formulate their models usually in the form of a Hamiltonian. In quantum mechanics, the Hamilton operator (Hamiltonian) is of fundamental importance in most formulations of quantum theory. Mentioned operator corresponds to the total energy of the system and its spectrum determines the set of possible outcomes when one measures the total energy. Interpretation of results obtained by the applying of models based on the Hamiltonian indicates very specific mechanisms of some observed phenomena that are not fully consistent with the experience. Such approach may occasionally lead to surprises when obtained results are confronted with expectations. The aim of this work is to find Hamilton operator of acoustic phonons in inhomogeneous solids. The transport of energy in the vibrating crystal consisting of ions whose properties differ over long distances is described in the work. We modeled crystal lattice by 1D “inhomogeneous” ionic chain vibrating by acoustic frequencies and found the Hamiltonian of such system in the second quantization. The influence of long-distance inhomogeneities on the acoustic phonons quantum states can be discussed on basis of our results.展开更多
We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within ...We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz,?where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.展开更多
In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct effi...In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear iterations.Various experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.展开更多
基金Supported by the National Natural Science Foundation of China
文摘An effective,hermitian hamiltonian is derived in amodel space. Its perturbation expressions to third order approximation are given.The correlation energy is also given to the third order approximation.The effective hamiltonian deviates form the actual one by the presence of acorrelation operator.The cprrelation operatop is given in an explicit form.
基金National Natural Science Foundation of China under grant No.10775097the President Foundation of the Chinese Academy of Sciences
文摘By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for someHamiltonians describing nonlinear processes in particle physics.In this way the energy-gap of the Hamiltonians can benaturally obtained.The characteristic polynomial theory has been fully employed in our derivation.
基金supported by the NSFC(Grants 12371403,12271426)by the Key Research and Development Projects of Shaanxi Province(Grant 2023-YBSF-399).
文摘In this paper,we formulate and analyse a kind of parareal-RKN algo-rithms with energy conservation for Hamiltonian systems.The proposed algorithms are constructed by using the ideas of parareal methods,Runge-Kutta-Nystr¨om(RKN)methods and projection methods.It is shown that the algorithms can exactly pre-serve the energy of Hamiltonian systems.Moreover,the convergence of the integra-tors is rigorously analysed.Three numerical experiments are carried out to support the theoretical results presented in this paper and show the numerical behaviour of the derived algorithms.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
基金Project supported by the National Natural Science Foundation of China (No. 11071067)the Hunan Graduate Student Science and Technology Innovation Project (No. CX2011B184)
文摘The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the continuous finite element method （CFEM） belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.
基金This project is supported by the National Key Planning Development Project for Basic tesearch(GrantNo.1999032801),the National Outstanding Youth Scientist Foundation of China(Grant No.49835109)and the Na-tional Natural Science Foundation of China(Grant
文摘Based on the principle of total energy conservation, we give two important algorithms, the total energy conservation algorithm and the symplectic algorithm, which are established for the spherical shallow water equations. Also, the relation between the two algorithms is analyzed and numerical tests show the efficiency of the algorithms.
基金the U.S. National Science Foundation under Grant Nos. 0140300 and 0500291the Southeastern Universities Research Association, the National Natural Science Foundation of China under Grant Nos. 10175031+1 种基金 10575047the LSU-LNNU Joint Research Program under Grant No. C164063
文摘A class of quasi-exact solutions of the Rabi Hamiltonian,which describes a two-level atom interacting witha single-mode radiation field via a dipole interaction without the rotating-wave approximation,are obtained by using awavefunction ansatz.Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and thefield frequency satisfy certain relations.As an example,the lowest exact energy level and the corresponding atom-fieldentanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes-Cummings andcounter-rotating cases of the Rabi Hamiltonian.
基金Project supported by the National Natural Science Foundation of China (No.10471038)
文摘By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.
文摘Theoretical solid-state physicists formulate their models usually in the form of a Hamiltonian. In quantum mechanics, the Hamilton operator (Hamiltonian) is of fundamental importance in most formulations of quantum theory. Mentioned operator corresponds to the total energy of the system and its spectrum determines the set of possible outcomes when one measures the total energy. Interpretation of results obtained by the applying of models based on the Hamiltonian indicates very specific mechanisms of some observed phenomena that are not fully consistent with the experience. Such approach may occasionally lead to surprises when obtained results are confronted with expectations. The aim of this work is to find Hamilton operator of acoustic phonons in inhomogeneous solids. The transport of energy in the vibrating crystal consisting of ions whose properties differ over long distances is described in the work. We modeled crystal lattice by 1D “inhomogeneous” ionic chain vibrating by acoustic frequencies and found the Hamiltonian of such system in the second quantization. The influence of long-distance inhomogeneities on the acoustic phonons quantum states can be discussed on basis of our results.
文摘We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz,?where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.
基金supported by the National Natural Science Foundation of China(Grant Nos.12171245,11971416,11971242,12301508)by the Natural Science Foundation of Henan Province(Grant No.222300420280)+1 种基金by the Natural Science Foundation of Hunan Province(Grant No.2023JJ40656)by the Scientific Research Fund of Xuchang University(Grant No.2024ZD010).
文摘In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear iterations.Various experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.
