Page 6 - Physlets and Open Source Physics for Quantum Mechanics:

Figure 6 shows the short-term time evolution of the probability density of a Gaussian
wave packet in an infinite well with walls at x = 0 and x = 1. The packet has an initial
momentum to the right and the images are shown at equal-time intervals. We are
supposed to imagine the motion of the packet by reading the images from left to right.
We notice that as the packet moves to the right its leading edge encounters the infinite
wall first and reflects to the left back towards the middle of the well.

There are several interesting features about this problem that are not depicted in the
image, however. While we get the general sense of what happens, we lose a lot of the
specific details by focusing on just one facet of the motion. This is especially apparent
at the collision with the wall. At the classical collision time, the classical momentum is
zero, yet in the quantum-mechanical case, the average momentum of the packet is
negative [21,22]. On the much longer revival time scale we are interested in visualizing
the fractional and exact quantum-mechanical revivals of the initial wave packet. One
can again look at the position-space probability density, but there is no guarantee that
by just looking we can recognize precisely when the revivals occur. In addition,
showing just one image lacks the ability to show how the change of initial conditions
affects the resulting dynamics.

Well-crafted computer simulations, however, can. The following OSP examples show
the time evolution of wave packet dynamics in the infinite square well. We focus on the
quantum ‘bounce’ and quantum-mechanical revivals and use the time dependence of
the expectation values [23], so-called quantum carpets [24] and, in the future, the joint
position-momentum quasi-probability density using the Wigner function [25] to uniquely
depict wave packet evolution in time. A brief tutorial on wave packet phenomena and
quantum-mechanical revivals [26-28] can be found in the Appendix.

Wave Packet Revivals: Simulations
For our initial Gaussian wave packets we use a variety of initial momentum values
corresponding to the central values of n 0 = 0, 10, 20, 40, and 80 (where p 0 = n 0π/L), so
that the ratio of classical periodicity to spreading time varies (for n 0 = 40, T cl/t 0 = 10/π,
and significant spreading can be seen even over half a classical period). For ease of
calculation, we also set 2m = L = ħ = 1, while allowing the user to vary x 0, p 0, and ∆x 0 =
b/√2. Using our simplifications, the revival time, T rev, becomes: T rev = 2n 0T cl = 2/π. For
the infinite square well, longer time scales are not present due to the purely quadratic
dependence of the energy eigenvalues. The revival time scale can clearly be much
larger than the classical period for wave packets characterized by n 0>>1 (giving T rev/T cl
= 2n 0, which for n 0 = 40 gives T rev/T cl = 80).

The program we use to study revivals, QMSuperposition, inputs the expansion
coefficients, c n, of a general quantum-mechanical superposition shown in Eq. (A2).
The program does so by separately inputting the real, Re[c n], and imaginary, Im[c n]
parts of the expansion coefficients, c n (which were separately calculated in
Mathematica). An XML file, such as the one shown in Figure 3, stores these expansion
coefficients and other appropriate initial conditions for the program.

The wave functions are then calculated either numerically for any user-defined potential
energy function, V(x), or calculated analytically for the special cases of the infinite
square well, the periodic infinite well, and the simple harmonic oscillator. Depending on
the analysis one wishes to perform, one of the following programs based on
QMSuperpositionApp (which itself only shows the wave function) can be chosen:

o QMSuperpositionProbabilityApp: adds a view of the probability density
o QMSuperpositionExpectationXApp: adds a view of the expectation
value of x