Stability of Finite Difference Solution of 1D Time-Dependent Schrodinger Wave Equation

The study of the stability of solutions to differential equations is a fundamental and ongoing area of research in mathematics and applied sciences with numerous applications, and it provides a framework for analysing the behaviour of dynamical systems and predicting their long-term behaviour. For a numerical solution to be useful it must be both consistent and stable, and such a solution can be said to be stable if small errors in the initial data or in the numerical approximation do not grow unbounded as the computations progresses. In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. It is demonstrated that the solution is conditionally stable for the explicit finite difference technique and unconditionally stable for the implicit finite difference methods using the numerical algorithm's matrix representation. We will utilize a 1D harmonic oscillator problem to demonstrate this behaviour.