**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 *a*_{n} 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* |
*a*_{n} |

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:

.

### Like this:

Like Loading...

*Related*

Tags: Friday Physics, physics, Quantum Mechanics

This entry was posted on May 30, 2008 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

June 6, 2008 at 1:14 am |

[…] 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 […]

September 18, 2011 at 1:12 pm |

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.