1. The problem statement, all variables and given/known data The Hamiltonian for a particle moving in a gravitational field and under no other influences is H = (P^2)/2m - mgZ where P is the momentum in the Z direction. 1. Find d<Z>/dt. 2. Solve the differential equation d<Z>/dt to obtain <Z>(t), that is, <Z> as a function of t, for the initial conditions <Z>(0) = h, <P>(0) = 0. Compare it to the classical expression Z(t) = (-gt^2)/2 + h 2. Relevant equations 3. The attempt at a solution Part 1. I solved the problem using the Ehrenfest theorem and expanded the commutator, but turned out the exact answer was on Wikipedia. That was disappointing but at least my final answer was right. http://en.wikipedia.org/wiki/Ehrenfest_theorem d<Z>/dt = 1/m * <P> Part 2. <P> is a definite integral and therefore is a constant. Integrate both sides. d<Z>/dt = <P>/m ∫d<Z> = ∫<P>dt/m <Z> - <Z(0)> = <P>/m ( t - t0), t0 = 0. <Z> - h = <P>t/m, <Z> = h + <P>t/m Dimensionally, this is correct. <Z> is meters, h is meters, <P> is meters*kilogram/second, then divide by m and times t, <P>t/m is meters. However, this looks nothing like the classical version and I'm wondering, does -gt/2 correspond to <P>/m? In addition, is <P> really, really a constant? If so then what is the use of the <P>(0) = 0 initial condition?