Asymptotically symplectic integrators of 3rd and 4th order for the numerical solution of the Schrödinger equation

Abstract

The time-independent Schrodinger equation is one of the basic equations of quantum mechanics. Its solutions are required in the studies of atomic and molecular structure and spectra, molecular dynamics, and quantum chemistry. The solution of the one-dimensional time-independent Schrodinger equation is considered by symplectic integrators. The Schrodinger equation is first transformed into a Hamiltonian canonical equation. This chapter presents several necessary results concerning symplectic methods and introduces the concept of asymptotic symplecticness. The concept of asymptotic symplecticness is introduced and asymptotically symplectic methods of orders 3 and 4 are developed. Numerical results are also obtained for the one-dimensional harmonic oscillator.