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 quantum mechanics study sessions. I went ahead and added the integrand for the square of x, used to calculate the standard deviation, to the graph above as well.
NOTE: Had I graphed the integrands first, I could have gotten out of doing the integral for the expectation value altogether. The areas to the left and right of 0 obviously cancel.
NOTE 2: The graphs above were created with Google Docs which is free. No pricey graphing packages required these days! No more excuses!
Have fun with it!