Xue-Shen Liu

Numerical solution of one-dimensional time-independent Schrödinger equation by using symplectic schemes

The symplectic schemes are extended to the solution of one-dimensional time-independent Schrodinger equation. The Schrodinger equation is first transformed into a Hamiltonian canonical equation by means of the Legendre transformation, and then two methods are developed to solve the numerical solution of the one-dimensional time-independent Schrodinger equation: the symplectic scheme-matrix eigenvalue method and the symplectic scheme-shooting method. Both methods are applied to the calculations of a one-dimensional harmonic oscillator, the hydrogen atom, and a double-well anharmonic oscillator. It is shown that the numerical results of the two methods are nearly the same and are in good agreement with the exact ones when the step length is taken to be properly small. The computation with the symplectic scheme-shooting method spends less computer time than that with the symplectic scheme-matrix eigenvalue method. And thus the symplectic scheme-shooting method is a better numerical method for the calculation of the eigenvalue problem.

The Symplectic Algorithm for Use in A Model of Laser Field

Using the asymptotic boundary condition the time‐dependent Schrodinger equations with initial conditions in the infinite space can be transformed into the problem with initial and boundary conditions, and it can further be discrected into the inhomogeneous canonic equations. The symplectic algorithms to solve the inhomogeneous canonic equations have been developed and adopted to compute the high‐order harmonics of one‐dimensional Hydrogen in the laser field. We noticed that there is saturation intensity for generating high‐order harmonics, which are agree with previous results, and there is a relationship between harmonics and bound state probabilities.

Gauss numerical inversion for use in computing the radiation field from a cylindrically symmetric radiation source

Abstract Abel inversion of a cylindrically symmetric radiation source involves the solution of the generalized Abel equations. In this paper, by using the Gauss numerical integral (GNI), the Abel equation can be separated into a system of linear algebraic equations, whose coefficient matrix is an upper triangle matrix and can be easily solved explicitly. Its numerical solution can achieve second-order accuracy. Three examples are evaluated using the present GNI. It is shown that the high accuracy is achieved and the computed values converge to the exact solutions with an increase in the number of nodes. This method is simple and requires less computer time, thus the experimenters can apply this method easily.