Numerical methods with a high order of accuracy applied in the quantum system

Abstract

Two kinds of numerical methods with a high order of accuracy are developed in this paper. In the general classical Hamiltonian system, it was claimed that no explicit n-step symplectic difference method with the 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 exploit to investigate the high order Newton–Cotes differential methods in the quantum system. For the first time, we work out the generalized derivation of explicit symplectic difference methods with any finite order of accuracy in the quantum system. We point out that different coefficients in the same multistep symplectic method will lead to quite different 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 symplectic difference structure, Newton–Cotes differential schemes, and other methods are presented. Numerical tests on the model system have also been carried out. The comparison shows that the explicit symplectic difference methods and the Newton–Cotes differential methods are both accurate and efficient.