The Schrödinger equation can be written , where is the mass of the ball (idealized as a point mass), is the acceleration of gravity, and is the vertical height (with ground level taken as ). For perfectly elastic collisions, the potential energy at can be assumed infinite: , leading to the boundary condition . Also, we should have as .
The problem, as stated, is not physically realistic on a quantum level, given Earth's value of , because would have to be much too small. But an analogous experiment with a charge in an electric field is possibly more accessible. We will continue to refer to the gravitational parameters, however.
Redefining the independent variable as , the equation reduces to the simpler form . (The form of the variable is suggested by running DSolve on the original equation). The solution that remains finite as is found to be . (A second solution, , diverges as .)
The eigenvalues can be found from the zeros of the Airy function: , using N[AiryAiZero[n]]. The roots lie on the negative real axis, the first few being approximately , , , , , , …. Defining the constant , the lowest eigenvalues are thus given by , , , and so on. The corresponding (unnormalized) eigenfunctions are . These are plotted on the graphic.
The semiclassical phase integral gives quite accurate values of the energies. Evaluate these using (the added fraction is , rather than the more common , because one turning point is impenetrable). The integral is explicitly given by , leading to . The first six numerical values are , compared with the corresponding exact results from the Schrödinger equation .
D. ter Haar, ed., Problems in Quantum Mechanics, 3rd ed., London: Pion Ltd., 1975 pp. 6, 98-105.