NUMERICAL-METHODS WITH A HIGH-ORDER OF ACCURACY APPLIED IN THE QUANTUM SYSTEM

Two kinds of numerical methods with a high order of accuracy are devel oped in this paper. In the general classical Hamiltonian system, it wa s claimed that no explicit n-step symplectic difference method with th e nth order of accuracy can be achieved if n is larger than 4. We show that there is no such constraint in the quantum system. We also explo it to investigate the high order Newton-Cotes differential methods in the quantum system. For the first time, we work out the generalized de rivation of explicit symplectic difference methods with any finite ord er of-accuracy in the quantum system. We point out that different coef ficients in the same multistep symplectic method will lead to quite di fferent results. The choices of coefficients and order of accuracy for the best efficiency in multistep symplectic methods and Newton-Cotes differential methods are studied. The connections between explicit sym plectic difference structure, Newton-Cotes differential schemes, and o ther methods are presented. Numerical tests on the model system have a lso been carried out. The comparison shows that the explicit symplecti c difference methods and the Newton-Cotes differential methods are bot h accurate and efficient. (C) 1996 American Institute of Physics.