Quantum Mechanics, Boundary Value Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg49525] Quantum Mechanics, Boundary Value Problem*From*: danese at physics.rutgers.edu (anthony danese)*Date*: Thu, 22 Jul 2004 02:46:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, I'm trying to find a numerical solution to Schrodinger's equation for a nearly free electron potential (V(x)=Cos(x)) which meets an image potential which is of the form (V(x)=C - 1/x) where C is a constant. I can set up the problem using NDSolve, with 2 of my boundary conditions being continuity of the wave function (psi) and its derivative(psi') where the 2 potentials meet. The problem is that I need 2 other boundary conditions--I attempt to arbitrarily set psi = 0 far into the Cos(x)-like region of the potential and I would like psi to vanish as for large values of x in the C-1/x region of space. When I try to run NDSolve, I either get the trivial psi = 0 or I get an error telling me that the boundary conditions are not well defined. As a test, I have tried to run NDSolve to find psi for a step potential where the step potential is greater than the energy of the electron. The solution should be free electron-like away from the step and the solution should decay into the step. This is a problem that is easily solved analytically but Mathematica doesn't like the way I'm stating the boundary conditions. Is there anyway to solve these particular examples of Schrodinger's equation with NDSolve? Any help would be greatly appreciated, anthony danese