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Phys 498CQM Lecture 2A

Author: Erik Koch, Modified by R. Martin 		HW 1 (due 1/31)

Outline

See Thijssen, Appendix; Koonin, Ch. 2,3

Ordinary Differential Equations

Much of physics is cast in terms of differential equations. ODE's can always be cast in the form

du/dx= F(u(x),x).

For example, a second order equation can be reduced to two coupled first order equations. The key numerical problem is to integrate the equations subject to Boundary conditions. Second-order equations having the general form

[ d2/dx2 + k(x) ] u(x) = S(x)

are especially important and present special issues: boundary conditions may be required at more than one point, so that these become boundary value problems (for example the Poisson equation with k(x) = 0) and the time independent Schrödinger equation for a bound state is an eigenvalue problem where k involves the unknown eigenvalue. These are of special interest for us.

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 is

[ (1/2) d2/dx2 - V(x) ] u(x) = E u(x)

If we let k2(x) = 2 [ 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 uj + 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( h2).

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 is O( h4 ), 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 can be rewritten using

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

to become

(1 + h2 k2j+1 / 12)uj+1 -2(1 - 5 h2 k2j / 12)uj + (1 + h2 k2j-1 / 12)uj-1 =0 + O(h6).

This can be solved (either forward or backward) to give uj+1 in terms of information at points j+1, j, and j-1, with no more computational complexity than the simple equation that has much lower accuracy. For the general equation with a source term S(x), the form becomes

(1 + h2 k2j+1 / 12)uj+1 -2(1 - 5 h2 k2j / 12)uj + (1 + h2 k2j-1 / 12)uj-1 =(h2 / 12)(Sj+1 + 10 Sj +Sj-1) + O(h6).

Last Modified Jan. 22
Email question/comments/corrections to rmartin@uiuc.edu