I got a quick lesson on the importance of graphing my work this afternoon. After diligently calculating the expectation position of a particle as predicted by the Schrodinger equation: I came up with the value 1/2λ. After a little bit of thought, it occurred to me that the Schrodinger equation was symmetric about 0 and that if I was guessing the expectation, or average, value for the position of a particle predicted by the equation, I'd guess zero. So, I graphed the Schrodinger equation for the potential. Just for good measure, I went ahead and graphed the integrand of the expectation integral as well. Sure enough, the Schrodinger equation is symmetric about 0 and the areas under the expectation integrand curve to the left and right of (x=0) cancel out. Upon checking my work I realized that I was missing a factor of x in my original integration. I multiplied the x back in and everything worked out just like the graph said it should! Not bad for taking a 20 year break between ...