Page 11 - Physlets and Open Source Physics for Quantum Mechanics:

Time Evolution of a Quantum Wave Packet on a Ring

Figure 12: A free-particle Gaussian wave packet confined to a ring visualized
with the same program used in Figure 9, but in this case the ‘ring’ is chosen for
the potential instead of the infinite square ‘well.’ The wave packet is shown at t
= 0, T cl/4, T rev/6, T rev/4, and T rev/2, respectively (T rev is again set to 1, but for the
ring, this is one quarter the revival time of the infinite square well). In the top
panels the wave function is shown in phase as color representation and in the
bottom panels the probability density is shown. The packet moves left to right
and leaves the right-hand side to re-enter on the left-hand side.

The final example we will discuss is that of a Gaussian wave packet confined on a
circular ring using the program QMSuperpositionProbabilityApp and the potential
energy function, V(x) set equal to “ring.” This is the same program used in Figure 9,
but now it uses periodic boundary condition. Here we visualize this example in two-
dimensions, but elsewhere we have used the new Open Source Physics Java3D library
[3,20] to show a three-dimensional depiction of the wave packet’s evolution. As shown
in Figure 12, the Gaussian wave packet evolves over time and undergoes quantum-
mechanical revivals. For this system, the wave packet again has an average
momentum, causing it to move around the ring, and again it spreads. After a short
time, the spreading is large enough that the front of the wave packet starts to interfere
with the back of the wave packet. At fractions of T rev, the revival time, the wave packet,
or ‘mini-packet clones’ of the original wave packet, can form.