Eigen Problem Of Schr(?)dinger Equation And Its Application In Photoionization

With the continuous development of laser technology in the past 60 years,its application has become more and more extensive,among which the interaction between strong laser and matter has become the focus of research.The interaction process can be described by Schr（?）dinger equation.The main solutions of Schr（?）dinger equation include split-operator method,generalized pseudo-spectral method,finite difference method,time-varying density functional method and backscattering method.In this paper,the characteristic problem of Schr（?）dinger equation under four typical potential fields is studied by using the finite difference method.Then,the numerical results which are approximate to the analytical solutions are obtained.On the basis of this,this paper studied the displacement of the spring oscillator after stretching,turned the practical problem into the harmonic oscillator evolution problem in the translational potential field,and used the Crank-Nicholson method（referred to as CN method）for numerical calculation,obtained reliable numerical results and verified the probabilistic property of the method.In addition,on the basis of solving the characteristic problem of soft-core potential field,the nonlinear scattering problem under the strong laser field(10¹²-10¹⁷W/cm²)is further studied.The time-dependent Schr（?）dinger equation was calculated by using the probabilistic Crank-Nicholson-like method（CN-like method for short）,and the harmonic spectrum and probability were obtained,which verified the probabilistic properties of the three-stage structure of harmonic radiation in physics.On this basis,the relationship between the scattering spectra and the parameters in the potential function,the laser field intensity and the space step size are studied.