In the code you typed why is there an r^2?

I just have one more question with taking the integral of the wave equations. When you take the integral of the product of two different wave equations you get the overlap correct? However is it just going to give zero or is it going to give me other functions describing the spaces the wave equations overlap

One of the weird things about quantum mechanics is that an electron can partially be in one state and partially in another state at the same time, like Schrodinger's cat. The overlap between two states of different energy is zero is because the mathematical operator which determines the states and the energies has certain symmetry properties.

The wave functions are not curves or surfaces. (technically, I mean that they are not 2D manifolds embedded in 3-space.) They are complex-valued scalar functions, which take on a value everywhere in space. They contain everything that can be known about the system they describe. In the full sense, they are time-dependent as well as varying in space.

But it is often meaningful to work with time-independent functions. These take on complex values everywhere. (They are continuous, so they can't just stop.) One of the things they determine is the probability of finding a particle in some region of space. This density function is given by Y* Y. (Where I use Y for psi, and * denotes the complex conjugate..) The probability of finding a particle in a region of space is given by the integral of Y* Y over that region.

So these are density functions. You can picture their value as a cloud in space of varying density. When these orbitals are plotted, it is this density function than is being represented. So, just for visualization, we plot some contour of constant density, but it is not a surface so far as the orbital is concerned. There is not much in the way of a precise meaning in the intersection of these contours. There is, however, a meaning in how much the orbitals "overlap" in all of space. Just like Y1* Y1 represents a density for Y1, Y1* Y2 represents the the extent to which the orbitals overlap, and the integral Y1* Y2 over all space represents the total overlap of the Y1 and Y2 orbitals.

So, in short, the wave functions are not an equation of the form f(x,y,z)=0, which could define a surface which could have an intersection with another surface. Rather, they are of the form f(x,y,z) -- not an equation -- where the values taken on vary from one wave function to another, and it is the way this variation in space occurs that contains information about the system.

The intersections are points where the different wave functions share a common coordinates. What I'm trying to do is find those common coordinates. and I'm having trouble reading the answers I got back.