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
.
Contributed by: S. M. Blinder (January 2012)
Open content licensed under CC BY-NC-SA
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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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