## 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:
.