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.
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