Computation Of The Eigenvalues Of The Schr(?)dinger Equation By Symplectic Partitioned Runge-Kutta Methods

The Schr(o|¨)dinger equation is one of the basic equations of quantum mechanics.For different potential fields,it is a second order differential equation which can be exactly solved only for a few typical potentials.Therefore,in practice the numerical methods must be used.Many authors have put forward a lot of approximate method,such asymptotic iteration method,1/N expansion method and mathematical software packages for solving this equation,to obtain numerical results of the eigenvalues and wave functions.These works have greatly promoted the continuous development of numerically solving Schr6dinger equation.On the other hand,it is well known that the Hamiltonian system can be used to describe the physical processes in the nature.The basic feature of the Hamilton system is its Symplectic property.Nevertheless,the numerical simulations of Hamiltonian systems using traditional numerical methods of ten destroy the Symplectic properties of the system trader consideration especially for the long time numerical simulation.Thus,it is natural and necessary to find a kind of Symplectic algorithms which can keep Symplectic structure of the Hamiltonian systems under consideration and attract a lot of scholar's concerns.Such a symplectic method has important practical application and theoretical significance.In this paper,we first review the basic concepts of quantum mechanics and Symplectic geometric.In the key chapter,the second one,we first give a short introduction to Symplectic algorithm and the existing Symplectic Partitioned Runge-Kutta methods on computing eigenvalues of the Schr6dinger equation.Secondly, We construct two new trigonometrically fitted Symplectic Partitioned Rung-Kutta format. In the third chapter,we apply our new Symplectic algorithm to numerically calculating eigenvalues of one-dimensional Schr(o|¨)dinger equation for harmonic oscillator potential field and Morse potential fields.We finally summarize the main results and some experience come from our practical calculation and outline a promising prospect for the further research work in the last chapter.All MATHEMATICA codes are given in this thesis are given in appendix...