Explicit Numerical Scheme for Solving 1D Schrödinger Equation

Once ReΨ and ImΨ are found from these two coupled di erential equations, the time-dependent probability density |Ψ| for a particle represented by the wave function Ψ can be calculated from |Ψ| = Ψ∗Ψ = (ReΨ) + (ImΨ), where the * denotes the complex conjugate. Our task, then, is to solve Equations (2) and (3) numerically, and this computational approach requires that the partial derivatives be approximated by something called nite-di erence equations. We will be interested, initially, in the propagation of a free particle, for which V (x) = 0. We will, however, develop a numerical scheme for solving the time-dependent Schrödinger Equation with a general nonzero potential energy V (x). The numerical solutions for a free particle can then be found by setting V (x) to zero in the resulting nite di erence equations.