Stability, convergence and error analysis of B-spline collocation with Crank-Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

In the present study, the complex-valued Schrodinger equation (CVSE) is solved numerically by a nonic B-spline finite element method (FEM) in quantum mechanics. The approach employed is based on the collocation of nonic B-splines over spatial finite elements, so that we have continuity of the dependent variable and its first eight derivatives throughout the solution range. For time discretization, the Crank-Nicolson scheme of second order based on FEM is employed. The method is shown to be unconditionally stable and accurate to order. Three problems are considered to validate the algorithm. Comparisons are made with existing methods and analytical solutions. The proposed method exhibits good conservation properties and performs well with regards to analytical solutions for different error norms and conservative constants related to parameters in quantum classes in Physics. The computational complexity of (2N+18) arithmetic operations with the help of a nonic-diagonal matrix is also tackled by the present scheme.