Calculation and Conservation of Probability and Energy in the Numerical Solution of the Schrödinger Equation With the Finite- Difference Time-Domain Method

The finite-difference time-domain (FDTD) method is a widely used numerical technique for solving Maxwell’s equations. FDTD has also been applied to numerically solve the Schrödinger equation and coupled systems of Maxwell and Schrödinger equations. In this work, we study the properties of the FDTD method for the Schrödinger equation, specifically the conservation of probability and energy. We propose accurate expressions for the total numerical probability and energy contained in a region and for the flux of probability current and power through its boundary. We show that the proposed expressions satisfy the conservation laws under suitable conditions and demonstrate their connection to the Courant–Friedrichs–Lewy stability limit. We discuss how these findings can be used to create new stable FDTD algorithms for the Schrödinger equation in an intuitive and modular fashion.