Computational Physics Group Meeting

Eric R. Bittner

26-May-2000

Numerov method for solving the time independent wave equation

Download this notebook: Numerov.nb

Preliminaries

The Numerov approach is a reasonably robust proceedure for solving the time independent Schrodinger equation (in 1 D). The proceedure works by specifying an initial value of ψ and its derivative  at a point and propagating on a spatial grid to some final point.   The basic equations of "motion" are derived by simply taking a Taylor series expansion about the point .  The estimated value of the function at some other nearby point, or is given by

Adding Eq. 1 and Eq. 2 and solving for f[n+1] yields to

Let f be a solution of the Schrodinger equation: (H - E) f = 0 .  Thus, we can obtain the second derivative of f using the operator equation

where

expanding f'', yields

where we have truncted the series at .  Now, solving for and substituting f'' = Gf, yields

which is our working equation.   Below are two examples of the Numerov method in action.

Harmonic Osc. States

Here we use the Numerov proceedure to try to find the eigenstates of a Harmonic well.  We know where the states should be a priori, so we can use this as a check of the proceedure.  Our proceedure is to pick an energy and propagate a solution.  A genuine eigenstate solution of this well will go to zero as x->±∞.  So, if we see the wavefunction solution blowing up, we know we do not have a solution at our choice of energy.  As it turns out, getting the exact solution is almost impossible.  So, we will try to zero in on the solutions by counting the number of nodes in the final wavefunction.  If as we increase the energy by one step we gain one additional node in the wave, we know that there must be a stable solution between the two choices of energy.  

The Schrodinger equation this case is

and we will use atomic units throughout (ℏ = 1, = 1).  

Example 2: H-Kr Scattering

Same proceedure as before, except in spherical coordinates.  The system here is H + Kr scattering with potential and parameters as given in Thijssen's book.  For scattering, the Numerov proceedure is a more preferable way to go about business since we only need to satisfy one boundary condition at r = 0.  As r -> ∞, we still need to satisfy scattering boundary conditions, however, this we can do by simply matching the log derivative of the Numerov solution to an asymptotic form of .  This is how we will compute the phase shifts and cross-sections.  

The Schrodinger equation we solve in this case is in radial coordinates:

Code:

Sample Calculations at various energies and total angular momentum.

Text and code copyright @ 2000 by Eric R. Bittner and University of Houston, all rights reserved. No part of this document may be used without the prior permission of the author.