I'm trying to solve numerically the 1-dim time dependent Schrodinger equation using the Crank Nicolson scheme and the Thomas algorithm to solve the tridiagonal matrix. The physical system consists of a free particle moving in a region of costant (zero) potential, far away from the boundary of my domain. At this time I am interested in checking the accuracy of the method experimentally, so I am studying the L2 error for different mesh spacing. The initial wave function and the analytical solution of the Scrhodinger eq. are the same that one can find in Cohen- Tannoudji (complement G1).
The problem is that I don't get the result I should (in particular, if I study the order of convergence experimetally I get inconsistent results and not the second order accuracy in space and time that I expect). Does anyone has any ideas? Thank you