虚时演化-劈裂算符方法在谐振子中的应用研究

Application study of virtual time evolution-split operator method to harmonic oscillator

摘要:

发展了一套以非微扰的方式求解含时薛定谔方程的理论方法(虚时演化-劈裂算符法)，该方法分别选用动量表象和坐标表象作为含时波函数演化的两个表象.在坐标表象下波函数的坐标部分使用库仑函数离散变量来离散，时间波函数展开在对应的格点上.以谐振子为例，进行了数值计算,发现在谐振势中放置两个电子，它们之间存在库仑相互作用.通过改变谐振势的强度，可以探索双电子基态波函数的定性变化.当谐振势较弱的时候，两个电子的波函数没有交叠，可作为Wigner晶格出现的证据.随着谐振势的增强，电子波函数开始发生重叠，类似于分子形成过程.

Abstract:

We present virtual time evolution-split operator method for solving the two-dimensional time- dependent Schrodinger equation. In this method, the Hamiltonian is accessed by employing the two representations of the wave function. One is a coordinate representations, in which the coordinate dependence of the wave function is discretized using a discrete variable constructed from the Coulomb wave function. Another is the momentum representation, the time function is expanded in the corresponding. As an example, the present method is applied to the harmonic oscillator. It is found that there exists coulomb interaction when two electrons are stored in the harmonic oscillator potential well. We explore the qualitative change of two electrons ground state wave function by altering the strength of the harmonic oscillator potential. When harmonic oscillator potential is weak, the wave functions are not overlapped, which can be used an evidence for presentence of Wingner lattice. With the increase of harmonic oscillator potential, the wave functions begin to overlap, which is similar to the mokecules formation.