This is a problem to a practice qualifying exam for graduate students. A series of marbles, each with mass m, is dropped from a height H directly above a line on the ground. Although a high precision dropping device is used, each marble does not land on the line. Show that the typical distance from the line where a marble lands is delta_x=(h_bar/m)^(1/2)*(2H/g)^(1/4) I'm thinking of the uncertainty principle delta_x*delta_p>=h_bar/2 but other than that, I don't know how to solve this problem. Trying to use the schrodinger equation to come up with <x>^2 and <x^2> in order to get (delta_x)^2 = <x^2> - <x>^2 seems pretty much impossible since the potential used here is gravitational and written in the form V = mg(H-x), so therefore to find the wavefunction, you have a differential equation with variable coefficients (which requires a complicated power series), and will take a lot of time. There should be an easy way to solve this since this is an exam problem.