Physics Friday 23: Quantum Bouncing Ball (Part 2/2)

Quantum Bouncing Ball Part 2: Probability Density and the Classical Limit.
(Part 1)

First, we note the spacing between the nodes of the wavefunction. As noted for large n, , meaning the spacing between and decreases toward zero as n increases without bound.

Also, with , we have nodes at all , , (and so ), for which (remember is negative), meaning that as the mass increases, the spacing between the nodes decreases; the spacing between fixed and is proportional to .

Now, we want to compare the high-energy probability distribution to the classical one. First, we calculate the classical probability distribution for bouncing ball. The probability of finding the ball in a particular interval is proportional to the fraction of time the ball spends in that interval. Working with the most basic repeating unit in the ball’s motion, a half bounce, we have the ball, with total energy E, moving from the height to 0 via the motion . Thus, the total time for a half cycle is . Thus, we have, with probability density function P(z):

Rearranging; and using the fact that, since dz and dt are both chosen to be positive, , where v is the velocity; we find:
Now, we have that , so . Combining with the position-time equation to find v(z), we obtain . Plugging in for and T, we find:

Now recall that as , the Airy function is approximated by , so with:
we find that, for ,
for .

Now, approximately averaging the rapidly varying sine square with , we get:

Which gives , as expected (recall that C is valued such that the integral of over all z is unity).

The upper asymptotic limit for is , meaning that P(z) drops to zero rapidly (faster that exponential) for .

Thus we approach the classical probability distribution when the energy is high ( with very high n).


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