Assignment #4

This assignment test your ability to perform numerical integration of single particle wave function. You have to integrate two Coulomb radial wave functions R_nl(r):

R₂₀(r) = (1/2a₀)^3/2 (2-r/a₀) * exp(-r/2a₀)

R₂₁(r) = 3^-1/2 (1/2a₀)^3/2 (r/a₀) * exp(-r/2a₀)

You are required to numerically evaluate the integrals:

I2;nl,n'l' = ó õ R 0 dr r2 Rnl Rn'l'

and

I4;nl,n'l' = ó õ R 0 dr r4 Rnl Rn'l'

for the following three combinations of quantum numbers nl, n'l':

n l = 20 ;	n'l'= 20
n l = 21 ;	n'l'= 21
n l = 20 ;	n'l'= 21

You can choose any value for the Bohr radius (including a₀=1), and the upper limit in the integral, R, should be greater then 20*a₀. You are required to use the best numerical integration method you know. You should use an appropriate number of mesh intervals and need to test the accuracy of your calculation.

Sample programs from the Numerical Recipes textbook for the trapezoidal integration (xtrapzd.f), optimized trapezoidal integration (xqtrap.f), Simpson integration (xqsimp.f), and Gaussian integration (xqgaus.f) can be found in the directory ~tomanek/PHY480/HW4/ on gauss.

The results should be submitted in a short report including the method you use, the accuracy you expect, the accuracy test you use, the FORTRAN code, etc. Demos of your code will be done after class.