Numerical Methods For Semi-classical Quantum Dynamical Systems

The Schrodinger equation is a fundamental equation of quantum mechanics. It combines material waves with the wave equations, and describes the motion state of the micro material in the form of probability distributions. The Schrodinger equa-tion is also a basic assumption of quantum mechanics, its correctness can be only verified by physical experiments. Semi-classical quantum mechanics is between the classical physics and the quantum physics, in fact, on the one hand it retains the classical mechanics theory about the trajectory of objects, on the other hand the physical observables in the Schrodinger equation also have the characteristics of probability distribution. From the viewpoint of the mathematical equations, the solution of the semi-classical Schrodinger equation is highly oscillatory in a local area, and this brings the trouble in the study about the mathematical theories and the numerical methods.Wc mainly consider three types of semi-classical problems, i.e. the semi-classical linear Schrodinger equation, the semi-classical nonlinear Schrodinger equa-tion in subcritical case and the semi-classical nonlinear Schrodinger equation in supercritical case. Using a diffeomorphism, we transform the original problems into the modified Schrodinger equations. Such transformation can reduce the os-cillatory frequency in the wave function, and slow down the movement of the wavepacket, thus this helps to solve numerically the semi-classical Schrodinger equations. Then, we give three kinds of numerical methods for solving the modi-fied equations derived by the three different semi-classical problems, respectively.For the semi-classical linear Schrodinger equation, in Chapter3, we present the improved Hagedorn wavepacket method, construct the simplified Hagedorn basis functions which are parameters-dependent, and give the spatial semi-discrete method of the modified Schrodinger equation. In the time-stepping computation, we make two improvements on the existing algorithms. On the one hand, we apply the skill of multi-time-step computation, i.e. computing the parametric equations with the small time stepsizes, and computing the coefficient equation with the large time stepsizes. The accuracy of the numerical solution can be higher for the smaller semi-classical scaling constant if we compute with different time stepsizes. On the other hand, based on the multi-time-step computation, we apply the Magnus expansion method to obtain the higher order accurate algorithm for solving the coefficient equation.In Chapter4, for the semi-classical nonlinear Schrodinger equation in sub-critical case, we still use the simplified Hagedorn basis functions in the spatial semi-discretization and the skill of the multi-time-step computation. Here, we combine such spatial semi-discretization with the time-splitting pseudo-spectral method. We give the collocation point and the discrete transform matrices, which are determined by the simplified Hagedorn basis functions. Then, we obtain the time-splitting wavepacket method.For the semi-classical nonlinear Schrodinger equation in supercritical case, the corresponding geometric properties are more complicated than the above two cases. Currently, there is no good way to resolve its analytical behaviors. In Chapter5, we construct the adaptive grids through the interpolating wavelet method, and use the time-splitting finite difference method on such wavelet-adaptive grids for solving the modified equations. From the numerical results, we can see that, the computational efficiency of the numerical method based on the wavelet-adaptive grids is greatly improved by comparison with the numerical method based on the equidistant grids.