| Schr(?)dinger differential equation and Heisenberg matrix algebra are two quite different mathematical formulations which open up the door of modern quantum mechanics.The standard mathematical formulations which provide the basis for calculations of quantum dynamics but are completely different with classical mechanics.In contrast,Feynman path integral is the third method of quantum mechanics,which builds on the familiar Lagrangian concept of the action of an orbit in space and time and appears to be much closer to classical concepts.In Feynman path integral formulation,the probability amplitude can be expressed by the coherent superposition of all possible spatio-temporal paths’ contributions from the initial state to the final state.Here,the weight of each path is a complex number,where the phase is equal to the classical action along the path.Although the time-dependent Schr(?)dinger equation(TDSE)can get quantitative agreement with the experimental results,it cannot give intuitive physical pictures.Moreover,numerically solving TDSE is only limited to a few simple atoms or molecules because of the complexity and difficulty of calculation.The statistical analysis method based on Feynman path integral provides an intuitive and quantitative explanation for the highly nonlinear processes with the view of classical Newtonian trajectories.Because these trajectories have been fully identified,the underlying physical mechanism can be traced according to the classical motion,which builds a bridge between the classical and quantum mechanics.We explore several statistical methods based on Feynman path integral,including Coulomb-corrected strong-field approximation,trajectory-based Coulomb-corrected strong-field approximation and Coulomb quantum-orbit strong-field approximation.The main contents,conclusions and innovations of our work are:(1)The detailed derivation processes of the above statistical analysis methods are introduced.The transition amplitude from bound state to continuous state is derived in coordinate space and momentum space,respectively.By employing the saddle point approximation,the integration of time is simplified to the summation form,and the initial conditions of the time-dependent dynamics of electrons are extracted.Genetic algorithm and Newton’s method are adopted to improve the efficiency of searching saddle points in complex plane.(2)We develop a new strategy named deep-learning-performed strong-field Feynman’s formulation,which allows us to overcome the shortcomings of mass data computation in existing methods.Our results show that the deep neural network can be trained effectively with a very small number of samples.Once trained,the method can predict directly the huge number of trajectories needed to reconstruct the highresolution spectra.In addition,the preclassification scheme is an effective method to tackle the trajectories’ diversity problem.Our method can not only overcome the challenges beyond the processing capacity of existing methods,but also open up the great potential of the highly efficient method in the analysis and prediction of strong field phenomena with deep learning.(3)A Coulomb-corrected strong-field statistical analysis algorithm based on Hartree-Fock wave function is proposed.The accurate atomic and molecular wavefunction can be obtained by using Gaussian quantum chemistry software,which overcomes the limitation of calculating only simple atoms by trajectory-based Coulomb-corrected strong-field approximation method.Based on above statistical analysis methods,the transition amplitude of electron is derived by molecular wavefunction,and the effect of Coulomb potential is further considered in the classical dynamics. |
