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Phys 498A Lecture Notes

Thursday, January 30, 1997
Lecturer: Erik Koch 		HW2 assigned

Outline

Numerical Integration of the 1-D Schrödinger Equation

The 1-D Schrödinger equation for an electron in a potential V(x) with units h = 1 and me = 1/2 is

[ - d2/dx2 + V(x) ] u(x) = E u(x)

If we let k2(x) = (2m/h) [ E - V(x) ], the Schrödinger equation takes the form

u''(x) = -k2(x) u(x).

We want to discretize this to a grid with spacing h. The second derivative operator can be discretized as

fj'' = ( fj+1 - 2 fj + fj-1 ) / h2 + O( h2 ),

where we have used the notation fj = f(xj) with the xj's being the grid points.

Direct application of the discritized derivative leads to a discretized Schrödinger equation with errors of order O( h4).

uj+1 - 2 uj + uj-1 = -k2j uj

This could be solved for uj+1 and used to integrate the equation. However, with a little extra work we can get a method that has errors of order O( h6 ), a substantial improvement known as the Numerov Method.

Numerov Trick

The error in the second derivative formula is from the fourth derivative of the function. We know this! By differentiating the Schödinger equation twice, we find that u''''(x) = -( k2 u )''(x). That is, knowledge about the curvature of the potential leads to a more accurate integration scheme.

The discretized 2nd derivative formula is

fj'' = ( fj+1 - 2 fj + fj-1 ) / h2 - h2/12 f(4)j + O( h4 ).

Thus the Schrödinger equation becomes

uj'' = ( uj+1 - 2 uj + uj-1 ) / h2 + ( k2j+1uj+1 - 2 k2juj + k2j-1uj-1 ) / 12 + O( h4 ).

Last Modified Jan. 30th
Email question/comments/corrections to shumway@uiuc.edu .