Exponential Functions Of Perturbative Series And Elimination Of Secular Divergences In Time-dependent Perturbation Theory In Quantum Mechanics

The development of time-dependent perturbation theory has shorter history than that of the stationary perturbation theory.Stationary perturbation theory originated from the perturbation method in astrophysics.This method was mature in the eighteenth century and it was the theoretical basis of the discovery of Neptune in 1846.In a simple system,however,the secular divergences difficulty in time-dependent perturbation theory had not been well solved until 1893 by Poincare(i.e.,the Poincare-Lindstedt method).For complex systems without a complete time-dependent perturbation theory,such as quantum mechanical systems,physicists have not come to realize that there were still many shortcomings in our understanding of the time-dependent perturbation theory in quantum mechanics once again until the discovery of Berry phase in 1984.In particular,the secular divergences difficulty has stimulated great research interests.Direct Polynomial Expansion Method is the idea of perturbation method in astrophysics,which is followed by the usual time-dependent perturbation theory in quantum mechanics.In contrast,the Poincare-Lindstedt method and geometric equivalence,etc.,suggest that exponential function should be utilized in the time-dependent perturbation theory in quantum mechanics,where the function of exponent is a polynomial expansion of perturbative series(Exponential Function of Perturbative Series).This thesis mainly consists of three parts.In the first part,we show the theoretical form of exponential function method for time-dependent perturbation theory of quantum mechanics,which includes the known geometric phase as a special case.In the second part,three exactly solvable models are used to verify the new method,which explain the rationality of this new theory,while the secular divergences exist in the traditional method.In the third part,as an application of the new method,we deal with a simple system without an exact solution.Divergent difficulty appears in the second-order term of this system,and in our theory it only contributes a phase factor.This study shows that secular divergences can be eliminated in the new method,while it is unavoidable in the traditional time-dependent perturbation theory.For finite discrete spectrum systems with a given initial state,the new method gives the same conclusion as a strictly solvable system.What is more,for those systems that cannot be solved strictly,this new method can also deal with the secular divergences appearing in the process of solving second-order approximation by traditional method.