``getPsi''

void getPsi(double Psi[], double E, int l,
            double V[], double r[], double dr[], int N)

Output:

• Psi[k]: values of normalized on all grid points r[k]

Input:

• E: electron energy found from ``getallEs''
• l: angular momentum of
• V[]: potential on grid points
• r[], dr[], N: standard grid points

Write a routine ``getPsi'', of the above prototype, which produces the values on the grid r[] of the wave function with energy E and angular momentum l, properly normalized so that . Your routine will consist of several steps:

First, you should construct the appropriate effective potential,

Next, you need to determine an appropriate point, kmatch, for matching the inward and outward solutions. To find the outermost point where V(r)<E, you may use the following fragment:

                        .
                        .
  kmatch=0;
  for (k=0; k<=N; k++)
    if (Veff[k]<E) kmatch=k;
  if (kmatch==0) {
    printf("Veff never below E=%f in getPsi.\n",E);
    exit(1);
  }
                        .
                        .

(This is the so-called ``classical turning point''.) Then, you should perform the inward (k=0..kmatch) and outward (k=N..kmatch) integrations, scaling both solutions to match so that .

Next, interpolate your result to include values at the odd points using your new routine ``interpolate'', and finally, normalize the result using the formula

where you perform the integral using your ``simpint'' routine.