Optional Problem: What is the Probability?

    Your job is to calculate the probability of finding the particle in the upper or lower lobe and in the middle section (doughnut).

    First, plot the orbital to find approximate values of the critical angles.   Second set the proper equation to find the critical values using the Solve function from MATHEMATICA.  Third, once you have found the critical angles, calculate the probabilities of the lobes and middle section. Finally, using the previous figures, plot only the lobe, and only the middle section of the orbital.
    

Integration in cartesian coordinates and three dimensions is given by

But in Spherical Coordinates, we have in general

In our case where we are not including the radial part

Remember that θ values range from 0 to π, and φ values from 0 to 2 π.

    Here is the integral of the square of the orbital