The "bouncing ball" revisited. Consider the quantum mechanical analog to the classical problem of a ball (mass m) bouncing elastically on the floor.
(a) What is the potential energy, as a function of height x above the floor? (For negative x, the potential is *infinite*—the ball can't get there at all.)
(b) Solve the Schrödinger equation for this potential, expressing your answer in terms of the appropriate Airy function (note that Bi(z) blows up for large z, and must therefore be rejected). Don't bother to normalize ψ(x).
(c) Using g=9.80 m/s2 and m=0.100 kg, find the first four allowed energies, in joules, correct to three significant digits. *Hint:* See Milton Abramowitz and Irene A. Stegun, *Handbook of Mathematical Functions*, Dover, New York (1970), page 478; the notation is defined on page 450.
(d) What is the ground state energy, in eV, of an *electron* in this gravitational field? How high off the ground is this electron, on the average? *Hint:* Use the virial theorem to determine ⟨x⟩.