It's a volume integral in spherical coordinates, so it has r^2 dr Sin[theta] dTheta dPhi. I didn't integrate over Theta and Phi because the r integral gave 0.

Back to Mathematica. Ignoring normalization constants, this shows that that the 1s and 2s orbitals have zero overlap:

In[1]:= psi1s[r_] = Exp[-r/a0];
psi2s[r_] = (2 - r/a0) Exp[-r/(2 a0)];

In[4]:= Integrate[psi1s[r]*psi2s[r] r^2, {r, 0, \[Infinity]}]

Out[4]= ConditionalExpression[0, Re[a0] > 0]

since a0 is a positive constant.

The absolute value squared of wave functions are probability densities. The overlap of the different wave functions of a system should be zero. This is necessary if they represent different energy states. Wave functions having the same energy are chosen so that their overlap is zero.

The wavefunctions are defined for all space so all coordinates are common. Perhaps you want the overlap, the integral of their product.

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

Ok now I think i'm starting to understand. The overlap has to be zero because an electron can't be in two different states at the same time.

I don't understand when you said "Wave functions having the same energy are chosen so their overlap is zero". Does that mean wave functions that have same energy levels have 0 probability to exist is one of two states (thinking of the Schrodinger's cat thought experiment). In that case how does phase play in all this. Isn't phase just changing the angle at which the system is orientated.

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.