# Quantum Mechanics of a Bouncing Ball

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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 .

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Contributed by: S. M. Blinder (January 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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 {2.32025,4.08181,5.51716,6.78445,7.94249,9.02137}, compared with the corresponding exact results from the Schrödinger equation {2.33811,4.08795,5.52056,6.78671,7.94413,9.02265}.

Reference

D. ter Haar, ed., *Problems in Quantum Mechanics*, 3rd ed., London: Pion Ltd., 1975 pp. 6, 98-105.

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