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 information

Write a routine getPsi() of the above prototype in ``physics.c'', 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. An appropriate value is the boundary between the exponential and oscillatory solutions, the so-called ``classical turning point'' where V(r)=E. 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 using schint(), scaling both solutions to match so that .

Next, interpolate your result to include values at the odd points using your new routine interpolate().

Finally, normalize the result using the formula

where you perform the integral using simpint().