Physics Friday 22: Quantum Bouncing Ball (Part 1/?)

Part 1: Setup and energy eigenvalues

Classically, we can describe a perfectly elastic vertically bouncing ball as the vertical motion under a constant gravitational force, with conserved energy, and an absolute barrier at z=0. To produce the quantum description, we need fo find the proper potential energy V(z) to plug into the Hamiltonian operator, and thus Schrödinger’s equation. As we have a perfect barrier at z=0, and the region z<0 is totally forbidden, we need V(z) to be infinite in that region. For z>0, the potential energy is simply mgz. Combining these, we obtain a triangular well:
.
Thus we have for z≤0, remembering that the wavefunction must be continuous at z=0.


Now, plugging this into the time-independent Schrödinger equation, we have

so for z≥0, we have:

Thus, we have the second-order linear differential equation for φ

Defining constants and , we can write this as:
.

Making the substitution gives us , which is Airy’s equation. The linearly independent solutions are the Airy functions. Normalization of the wavefunction requires that as , so we only have the Airy function Ai(u). Thus we see that the equation has solution:
for z>0, where C is a constant such as to normalize the wavefunction (that is, so that ). Plugging in the definitions of λ and k, we obtain:


But we also have the requirement that when z=0. This means that we only have certain allowed energy eigenvalues , such that
, or thus , where an is the nth zero of the Airy function Ai (in decending order, as they are negative). Approximate values of the first few zeroes are as follows:

Approximate values of the zeros of Ai(x)
n an
1 -2.33811
2 -4.08795
3 -5.52056
4 -6.78671
5 -7.94413


As , , which tells us that for large n,
, and thus the higher energy levels can be approximated as:
.

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2 Responses to “Physics Friday 22: Quantum Bouncing Ball (Part 1/?)”

  1. Physics Friday 23: Quantum Bouncing Ball (Part 2/2) « Twisted One 151’s Weblog Says:

    […] 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 […]

  2. Lester D. Hulett Says:

    I have looked far and wide on the internet to find a layman’s derivation of the Airy solutions for the bouncing ball. This is the best I’ve seen. I assume you have seen the Nesvixhevski experiments on the bouncing neutron that show that the neutron’s vertical motion is quantized. Let us hear your comments on this. I think this experiment will take its place in the basic physics curricula. Your quick and easy derivation of the Airy function will make neat supplement in the basic physics books.

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