Thank you very much for your help, but I'm a little uncertain what you mean by
This approximation results in the term E = (1/2m)p^2 + mgy, where y = H-x^2/{2H}, which I can't write in the form Bx^2. Is the constant term is not relevant?
"Consider an infinitely sharp pin of mass M and height H perfectly balanced on its tip. Assume that the mass of the pin is all at the ball on the top of the pin. Classically, we expect the pin to remain in this state forever. Quantum mechanics, however, predicts that the pin will fall over...