Entangled Trajectory Molecular Dynamics In Quantum Phase Space

Since the birth of quantum mechanics, people have been looking for the corre-spondence between quantum mechanics and classical mechanics. This forms semi-classical theory and the theory of quantum phase space. The semi-classical theory aims at studying the microscopic quantum systems with the concepts and methods from classical theory. Currently it has enjoyed a renaissance in many branches of physics, mainly because the quantum theory encountered new challenges in dealing with the problems of molecular systems. It is very complex to determine how to truncate huge and even infinite Hamiltonian and diagonalize it in a large set of basis functions due to factors like high dimension, nonlinearity and strong correlation. Besides, it is hardly to get a vivid physical image to describe the nature of the system dynamics, even with the numerical results. The picture of quantum phase space precisely provides a bridge between the quantum and the classical world. In1932Wigner introduced the quantum mechanical phase space picture and proposed the concept of quantum phase space distribution function. The Wigner function can only be considered as a quasi-probability distribution function because it can assume negative values, even for non-negative initial conditions. In quantum phase space we can conveniently discuss some properties of the correspondence between quantum and classical mechanics, and show the quantum effects. Quantum phase space distribution function and the wave function are equivalent to describe the state of quantum mechanics, and this allows one to describe the quantum system with classical language as much as possible.Quantum phase space theory can help us better understand the quantum world, and can offer semiclassical pictures for quantum effects. It has been widely applied in many fields, such as quantum optics, statistical physics, collision theory and non-linear physics. In quantum optics the density operator is used to define a higher order correlation functions and to discuss the quantum theory of optical coherence. In collision theory the movement of the particles has been investigated in the infi- nite potential well and the steps potential. And this theory involves calculating the reaction probabilities between helium atoms and hydrogen molecules, and between hydrogen atoms and hydrogen molecules. The application of the quantum phase space mainly focuses on the following two aspects:as an effective computational tools; establishment of the trajectory equations of motion to simulate complex quan-tum process according to the quasi-probability distribution function theory. The significance of the quantum phase space is that an appropriate quantum phase space distribution function can be chosen to describe the quantum system with intuitive classical mechanics image. The entangled trajectory molecular dynamics (ETMD) formalism in this paper gives a unique and appealing physical picture of the quan-tum tunneling process. The motion of the quantum trajectories is entangled with each other and the trajectory ensemble must be propagated as a unified whole due to the nonlocality of quantum states. Trajectories for initial energy below the barrier with time evolution can successfully escape the metastable well do so by "borrowing" enough energy from their fellow ensemble members to surmount the barrier.By direct numerical methods for solving the time-dependent Schrodinger equa-tion, we can obtain the most accurate and comprehensive information about system dynamics. For complex systems, it is unfeasible to solve the Schrodinger equation, because of their exponential scaling with the increase of system size, while the methods of classical molecular dynamics can often be applied effectively to model complex molecular system. However, classical dynamics can not reproduce the es-sential physics of systems when quantum effects are significant. In recent years, combined with classical and quantum property of systems, the quantum trajec-tory method gained attention as an alternative way of solving the time-dependent Schrodinger equation, such as quantum hydrodynamic trajectory, the Wigner dis-tribution function and Husimi distribution function trajectory method. Since these methods are based on trajectories rather than grid points and can avoid the scaling bottleneck, they are highly desirable for multi-dimensional problems of molecular dynamics.In this thesis, the molecular dynamics of systems have been studied with the entangled trajectory molecular dynamic method. Molecular dynamics, as a new interdisciplinary subject connecting physics and chemistry, based on the theory of modern physics (especially in atomic and molecular physics, laser physics) and experimental technology (molecular beam, laser and computer technology), focuses on the movement and interaction of molecules and interactions. The motion of atoms is assumed to follow some definite description such as Newton equation, Lagrange equation or Hamiltonian equation, etc., which means the motion of the atoms relates to determined trajectory. In quantum systems, initially localized wave packets will spread and disperse with time evolution, which makes the wave packet dynamics of real system very complex. The wave packet dynamics can be well described by the autocorrelation function, which is a projection of the wave function at time t into its initial state and can be measured in experiment.Since2001people develop the ETMD method, it has been successfully applied to many one-dimensional systems, like the reaction probability, tunneling rate. In the past decades, ETMD method has got wide applications and the theory itself has made constant improvement and development. For example, it has been applied to solve the phase space diffusion equations, simulate of quantum processes using Wigner function, and develop entangled trajectory dynamics in the Husimi repre-sentation. In particular, our research members develop the integrodifferential form of evolution equation for the Wigner function, without a Taylor series expansion of potential. We have mainly done the following work:1. The autocorrelation function is investigated in the formalism of entangled trajectory molecular dynamics. The results of ETMD are consistent well with quantum simulations. We investigate the contribution of individual trajectory member of the trajectory ensemble to the autocorrelation function, and show the vivid physical picture of the autocorrelation via individual trajectory. Also, we find the "autocorrelation" of trapped trajectory is higher than that of the escaped trajectory.2. We extend the ETMD method to multidimensional systems, and apply it to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum simulations. By the comparison of the quantum and classical trajectory in phase space, the ETMD method can show an intuitive physical picture of the quantum tunneling phenomenon.3. We extend the ETMD method to simple chemical reaction process, and cal-culate the reaction probability of the collinear H+H2reaction. The result shows it is consistent with quantum simulations.This thesis consists of seven chapters:Chapter Ⅰ is the introduction of existed three quantum trajectory methods: the quantum hydrodynamic trajectory method, Wigner distribution function and Husimi distribution function trajectory method. Then, the authors introduce the motion of the quantum trajectories and discuss some basic properties about these methods.In chapter Ⅱ, some basic theoretical methods are provided. Firstly, it is about the definition of quantum phase space distribution function, and the introduction of four distribution functions such as Wigner distribution function, Husimi distri-bution function, standard (or anti-standard) and nomal (or anti-nomal) ordering distribution function. Then, some basic concepts of mathematical statistics, as well as several effectively used density estimation methods are introduced. The end of this chapter is the detailed description of dynamics in phase space, and derivation of the entangled trajectory molecular dynamics formalism.In the chapter Ⅲ, we derive the quantum entangled trajectories equations in detail.In the chapter Ⅳ, based on entangled trajectory molecular dynamics method, calculation of the autocorrelation function of three one-dimensional model systems are showed, with the comparation of the exact quantum mechanical results. We investigate the contribution of individual trajectory member of the trajectory en-semble to the autocorrelation function, and show the vivid physical picture of the autocorrelation via individual trajectory.In the chapter Ⅴ, we extend the entangled trajectory molecular dynamics method to multidimensional systems, and derive the trajectory evolution function in detail. And then the method is applied to a two-dimensional model, the results show that it is well consistent with exact quantum simulations. The entangled trajectory molec-ular dynamics method gives an intuitive physical picture of the quantum tunneling process. Finally, we discuss the relationship between computational effort in the ETMD calculation and the dimensionality of the system.In the chapter Ⅵ, we investigate the one-dimensional and two-dimensional of collinear H+H2model, calculate the reaction probability as a function of initial wavepacket energy. The conclusion has been drawn that although the results of entangled trajectories molecular dynamics are not in good agreement with quantum mechanics simulations, the numerical trend is consistent with each other.Chapter Ⅶ is about the summary of the study and the proposal of a new entangled trajectory molecular dynamics method which contains negative value, and gives a further development and application of the entangled trajectory molecular dynamics formalism.