Square-Preserving and Symplectic Structure and Scheme for Quantum System

The time-dependent Schrödinger equation is a square-preserving and symplectic (SPS) transformation. The canonical equations of quantum systems are deduced by using eigenfunction expansion. The normal-square of wavefunction of the quantum systems is an invariant integral of the canonical equations and then the symplectic schemes that based on both Cayley transformation and diagonal Padé approximation to exp(x) are also square-preserving. The evaluated example show that the SPS approach is reasonable and effective for solving time-evolution of quantum system.