BohrRadius.java

Below is the syntax highlighted version of BohrRadius.java from §1.3 Conditionals and Loops.

/*************************************************************************
 *  Compilation:  javac BohrRadius.java
 *  Execution:    java BohrRadius r
 *  
 *  Find the radii where the probability of finding the electron in the
 *  4s excited state of hydrogen is zero. Te probability is given by:
 *
 *      (1 - 3 r / 4 + r^2 / 8 - r^3 / 192)^2 exp(-r/2)
 *
 *  where r is the radius in units of the Bohr radius (0.529173E-8 cm).
 *  This is equivalent to finding the roots of
 *
 *       f(r) = 1 - 3r/4 + r^2 / 8 - r^3 / 192 = 0
 *
 *  We find them using Newton's method. The true roots are:
 *  1.871644455, 6.610814579, and 15.51754097. Depending on which
 *  value we use to start Newton's method, we find a different one.
 *
 *
 *  % java BohrRadius 0.0
 *  f(1.8716444550481757) = 1.1796119636642288E-16
 *
 *  % java BohrRadius 10
 *  f(6.610814578664557) = 1.1102230246251565E-15
 *
 *  % java BohrRadius 13
 *  f(15.517540966287266) = -3.552713678800501E-15
 *
 *  % java BohrRadius 20
 *  f(15.517540966287264) = 0.0
 *
 *************************************************************************/

public class BohrRadius {

   public static void main(String[] args) {
      double r = Double.parseDouble(args[0]);

      double EPSILON = 1E-14;         // desired accuracy
      double f  = 0.0;                // f(r)
      double df = 0.0;                // f'(r)

      // do at most 20 Newton iterations
      for (int i = 0; i < 20; i++) {

         // the function f(r)
         f = 1 - 3*r/4 + r*r/8 - r*r*r/192;

         // the derivative f'(r)
         df = -3.0/4.0 + r/4 - r*r/64;
         
         // Newton update
         r = r - f / df;

         // stop early if found root up to desired accuracy
         if (Math.abs(f) < EPSILON) break;

      }

      System.out.println("f(" + r + ") = " +f);
   }
}