Ding Pei-Zhu

Square-Preserving and Symplectic Structure and Scheme for Quantum System

The time-dependent Schrodinger equation is a square-preserving and symplectic (SPS) transformation. The canonical equations of quantum systems are deduced by using eigenfunction expansion. The normal-square of wavefunction of the quantum systems is an invariant integral of the canonical equations and then the symplectic schemes that based on both Cayley transformation and diagonal Pade approximation to exp(x) are also square-preserving. The evaluated example show that the SPS approach is reasonable and effective for solving time-evolution of quantum system.

Calculations of Photo-Ionization Cross Sections for Lithium Atoms

Photo-ionization cross sections for the ground and the n ≤ 5 excited states of lithium atoms are calculated in the photoelectron energy ranging from threshold to 0.5 Rydberg. The wavefunctions for both the bound and continuum states are obtained by solving the Schrodinger equation numerically in a symplectic scheme. Our results are in excellent agreement with the recent experimental measurements and in harmony with other theoretical calculations.

Dynamic properties of the cubic nonlinear Schrödinger equation by symplectic method

The dynamic properties of a cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method with different space approximations. The behaviours of the cubic nonlinear Schrodinger equation are discussed with different cubic nonlinear parameters in the harmonically modulated initial condition. We show that the conserved quantities will be preserved for long-time computation but the system will exhibit different dynamic behaviours in space difference approximation for the strong cubic nonlinearity.

Symplectic Schemes and the Shooting Method for Eigenvalues of the Schr dinger Equation

The one-dimensional time-independent Schrodinger equation is transformed into a Hamiltonian canonical equation by means of the Legendre transformation, then the symplectic schemes and a new shooting method extended to the eigenvalues of the Schrodinger equation. The method is applied to the calculations of one-dimensional harmonic oscillator, an anharmonic oscillator and the hydrogen atom. The numerical results are in good agreement with the exact ones.

An interpolation solution of the Abel transformation for use in optically-thick, cylindrically-symmetric plasmas

Abstract An interpolation to obtain the radial emission coefficient for optically-thick, cylindrically-symmetric plasmas is proposed. Using the interpolation, the generalized Abel equation may be separated into a system of linear algebraic equations. The coefficient matrix of the system is an upper Heisenberg matrix. An example is evaluated using both our interpolation and Youngs iteration. It is shown that the interpolation is precise and easier to apply than Youngs iteration.

Behaviour of Cubic Nonlinear Schr?dinger Equation by Using the Symplectic Method

The dynamic properties of cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the trajectories in phase space will exhibit different behaviour (elliptic orbit or homoclinic orbit) with the increase of nonlinear perturbation. We illustrate this phenomenon by mean of linearized stability analysis. The theoretical analysis is consistent with the numerical results.

Periodic and Chaotic Breathers in the Nonlinear Schr?dinger Equation

The breathers in the cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the solitonlike wave, the periodic, quasiperiodic and chaotic breathers can be observed with the increase of cubic nonlinear perturbation. Finally, we discuss the breathers in the cubic-quintic nonlinear Schrodinger equation with the increase of quintic nonlinear perturbation.

Existence of boundary value problems for TFD equation in quantum mechanics

TFD (Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lower solution method.