Variational-Iterative Solution of Ground State for Central Potential

The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials. As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.     

Variational-Iterative Solution of Ground State for Central Potential

The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials.As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.     

A hybrid discontinuous Galerkin method for computing the ground state solution of Bose–Einstein condensates

A numerical method for computing the ground state solution of Bose–Einstein condensates modeled by the Gross–Pitaevskii equation is presented. In this method, the three-dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions. Both polynomial and plane wave bases are considered for this purpose, and Lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution. The ground state is computed by an iterative procedure for minimizing the energy. The performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher-order finite element method.     

Studies of the Ferromagnetic State in Quasi-One-Dimensional Organic Polymer Ferromagnets

The possible ferromagnetic ground state of π-conjugated alternant hydrocarbon chains with unequal number of atoms on two sublattices is studied by the peierls-Hubbard model. Allowing for full lattice relaxation, a set of self-consistent iterative equations are established to study this system. The stabilities of the possible ferromagnetic ground states with respect to two kinds of unsymmetric site energies are also studied. We find that the ground state of the system is a stable ferromagnetic state as a result of the topological structure of the network and the electron-electron interaction. And the unsymmetric site energies, induced by substitutions, will severely weaken the stability of the ferromagnetic ground state.     

Accurate determination of tensor network state of quantum lattice models in two dimensions

We have proposed a novel numerical method to calculate accurately physical quantities of the ground state using the tensor network wave function in two dimensions. The tensor network wave function is determined by an iterative projection approach which uses the Trotter-Suzuki decomposition formula of quantum operators and the singular value decomposition of matrix. The norm of the wave function and the expectation value of a physical observable are evaluated by a coarse-grain tensor renormalization group approach. Our method allows a tensor network wave function with a high bond degree of freedom (such as D=8) to be handled accurately and efficiently in the thermodynamic limit. For the Heisenberg model on a honeycomb lattice, our results for the ground state energy and the staggered magnetization agree well with those obtained by the quantum Monte Carlo and other approaches.     

Nonlinear optical rectification of hydrogenic impurity in a disk-like parabolic quantum dot: The role of applied magnetic field

We report a detailed theoretical study of the effect of combined electric and magnetic field on the nonlinear optical rectification of a hydrogenic impurity, confined in a two dimensional disk-like quantum dot, with parabolic confinement potential. We use the compact density matrix formalism and iterative method to obtain nonlinear optical rectification and absorption coefficients. To find energy levels and wave functions, we employ exact diagonalization method in the effective mass approximation. As main result, we found that the transition energy from ground to first excited state redshifts with increasing the magnetic field while blueshifts for transition from ground to second excited state, moreover, for former transition, nonlinear optical rectification coefficient decreases with increasing magnetic field in contrast to that occurs for latter one.     

Characterization of an optical frequency comb using modified direct frequency comb spectroscopy

We introduce a method for determination of the absolute frequencies of comb lines within an optical frequency comb spectrum. The method utilizes the experimental and theoretical approach of the velocity-selective optical pumping of the atomic ground state hyperfine levels induced by resonant pulse-train excitation. The information on the laser pulse repetition frequency and carrier–envelope offset are physically mapped onto the ⁸⁷Rb ground state hyperfine level population velocity distributions. Theoretical spectra are calculated using an iterative analytic solution of the optical Bloch equations describing the resonant pulse-train excitation of four-level ⁸⁷Rb atoms. They are employed to fit the measured spectra and obtain the parameters of the frequency comb, thus providing a practical algorithm which can be used in real-time measurements.     

量子激发态最陡下降微扰理论

本文发展了量子激发态能量与波函数的最陡下降微扰理论计算方法,该方法避免了普通微扰理论所需要的对于参考态的无限求和困难,并能通过逐步迭代计算逼近于体系精确的本征函数和本征值。只要保持激发态试探波函数正交于其对称性相同的低激发态或基态的波函数,避免计算过程中的变分坍陷,本文的方法能用于求精确的激发态能量和波函数。 关键词：     

Coulomb Scattering in the Massless Nelson Model I. Foundations of Two-Electron Scattering

We construct two-electron scattering states and verify their tensor product structure in the infrared-regular massless Nelson model. The proof follows the lines of Haag-Ruelle scattering theory: Scattering state approximants are defined with the help of two time-dependent renormalized creation operators of the electrons acting on the vacuum. They depend on the ground state wave functions of the (single-electron) fiber Hamiltonians with infrared cut-off. The convergence of these approximants as t→∞ is shown with the help of Cook’s method combined with a non-stationary phase argument. The removal of the infrared cut-off in the limit t→∞ requires sharp estimates on the derivatives of these ground state wave functions w.r.t. electron and photon momenta, with mild dependence on the infrared cut-off. These key estimates, which carry information about the localization of the electrons in space, are obtained in a companion paper with the help of iterative analytic perturbation theory. Our results hold in the weak coupling regime.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential: Excited states

The quasilinearization method (QLM) is used to approximate analytically, both the ground state and the excited state solutions of the Schrödinger equation for arbitrary potentials. The procedure of approximation was demonstrated on examples of a few often used physical potentials such as the quartic anharmonic oscillator, the Yukawa and the spiked harmonic oscillator potentials. The accurate analytic expressions for the ground and excited state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first QLM iteration. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. The choice of zero iteration is based on general features of solutions near the boundaries. In order to estimate the accuracy of the QLM solutions, the exact numerical solutions were found as well. The first QLM iterate given by analytic expression allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems.     

Helium trimer calculations with a public quantum three-body code

We present an illustration of using a quantumthree-body code being prepared for public release. The code is based on iterative solving of the three-dimensional Faddeev equations. The code is easy to use and allows users to perform highly-accurate calculations of quantum three-body systems. The previously known results for He₃ ground state are well reproduced by the code.     

Analysis of the SU(3) model in a multistep variational approach

We discuss a multistep variational approach to collective excitations. The approach is developed in a boson formalism (bosons representing particle-hole excitations) and based on an iterative sequence of diagonalizations in subspaces of the full boson space. The purpose of these diagonalizations is that of searchingf or the best approximation of the ground state of the system. The procedure also leads us to define a set of excited states and, at the same time, of operators which generate these states as a result of their action on the ground state. We examine the cases in which these operators carry one-particle-one-hole and up to two-particle-two-hole excitations. We also explore the possibility of associating bosons to Tamm-Dancoff excitations and of describing the spectrum in terms of only a selected group of these. Tests within an exactly solvable three-level model are provided.     

The boson gas on a Cayley tree

We analyze the free boson gas on a Cayley tree using two alternative methods. The spectrum of the lattice Laplacian on a finite tree is obtained using a direct iterative method for solving the associated characteristic equation and also using a random walk representation for the corresponding fermion lattice gas. The existence of the thermodynamic limit for the pressure of the boson lattice gas is proven and it is shown that the model exhibits boson condensation into the ground state. The random walk representation is also used to derive an expression for the Bethe approximation to the infinite-volume spectrum. This spectrum turns out to be continuous instead of a dense point spectrum, but there is still boson condensation in this approximation.     

A new variational perturbation method for double-well oscillators

We propose a variational perturbation method based on the observation that eigenvalues of each parity sector of both the anharmonic and double-well oscillators are approximately equi-distanced. The generalized deformed algebra satisfied by the invariant operators of the systems provides well defined Hilbert spaces to both of the oscillators. There appears a natural expansion parameter defined by the ratio of length scales of the trial wavefunctions. The energies of the ground state and the first order excited state, in the zeroth order variational approximation, are obtained with errors <10⁻²% for vast range of the coupling strength for both oscillators. An iterative formula is presented which perturbatively generates higher order corrections from the lower order invariant operators and the first order correction is explicitly given.     

Multistep variational approach to collective excitations

We discuss a multistep variational approach to collective excitations. The approach is developed in a boson formalism (bosons representing particle-hole excitations) and based on an iterative sequence of diagonalizations in subspaces of the full boson space. Purpose of these diagonalizations is that of searching for the best approximation of the ground state of the system. The procedure also leads us to define a set of excited states and, at the same time, of operators which generate these states as a result of their action on the ground state. We examine the cases in which these operators carry one-particle one-hole and up to two-particle two-hole excitations. We also explore the possibility of associating bosons to Tamm-Dancoff excitations and of describing the spectrum in terms of only a selected group of these. Tests within an exactly solvable three-level model are provided.     

Rigorous evaluation of the ring diagram contribution to the ground state energy of an electron gas

The ground state energy and the correlation energy of an electron gas are evaluated rigorously without using the smallr _s expansion and the small momentum-transfer approximation in the ring diagram contribution and taking into consideration the first order and second order exchange graphs. The Fermi momentum is determined by solving the number density equation without using iteration and is compared with that obtained by iteration. The ground state energy is found to stay positive in contrast to the iterative solution which becomes negative beyond a certain value ofr _s .     

Formation of ground and excited hydrogen atoms in proton–potassium inelastic scattering

The inelastic scattering of proton with a potassium atom is treated for the first time as a three-channel problem within the framework of the improved coupled static approximation by assuming that the ground (1s state) and the excited (2s state) hydrogen formation channels are open for seven values of total angular momentum, ? (0≤?≤6) at energies between 50 and 500 keV. The Lipmann–Swinger equation and the Green’s function iterative numerical method are used to calculate iterative partial and total cross-sections. This can be done by calculating the reactance matrix at different values of the considered incident energies to obtain the transition matrix that gives partial and total cross-sections. Present results are in reasonable agreement with previous results.     

一种确定双原子分子基电子态的新方法

针对某些双原子分子基电子态与激发态势能曲线存在相交,从而引起利用从头计算确定分子基电子态时因初始输入不同而确定出的基态不同的情况,提出了一种将势能曲线计算和优化计算相结合确定基电子态的方法,并运用它确定B+2分子的基电子态.解释了文献中对于它的基电子态的计算出现不同结果的原因.     

Simple procedure for solving few-body equations

In this work we propose a procedure for iterative improvement of the adiabatic solution of the hyperspherical-harmonics technique. The usefulness of the method is demonstrated by calculating the binding energies of the ground and first excited state of the⁴He nucleus using several nucleon-nucleon interactions. The procedure is found to be simple and very stable. We compare the elastic charge form factor of the⁴He nucleus obtained by the usual adiabatic solutions and the improved version of this work. This comparison gives valuable physical insight into the nature of the adiabatic approximations.     

库仑加线性位基态解析解

应用新发展的单一轨迹积分方法求解库仑加线性位的基态量子波函数,得到基态能量和波函数的一般解析表达式,并讨论了解的收敛性.应用此方法讨论了重夸克偶素系统.