# Order of convergence of Scrodinger eq. with CN scheme

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 Matlab code is here: http://tny.cz/1bd1bf94. and the code for the Thomas algorithm (already tested) here: http://tny.cz/83c04868.

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

• How are you designing your tests, in particular, how do you choose time and space step size? Also, what are the orders you observe? – Guido Kanschat Jul 26 '13 at 20:46
• For a mesh spacing of 0.1 I get an error of 0.816. for a mesh spacing of 0.05 I get an error of 0.814 and for 0.025 mesh the error is 0.812.. now I have correct the code adding the boundary condition psi(x,t)=0 on the boundary (and put the wave number k=0 for simplicity), but the results are exactly the same. – the_elder Jul 27 '13 at 9:42
• What happens if you change time step size instead of spacial accuracy? – Guido Kanschat Jul 27 '13 at 11:45