Wave Propagation in Linear Media | re-examined

Wave functions in graphical representation
past. It is interesting to notice that they were chie y restricted to quantum mechanical wave
packets. Seemingly, the imagination of quantum particles moving like waves was much more
demanding and called for a visualisation rather than the analogous problem of electromagnetic
wave propagation. There are quite a number of examples of computer animation of wave
packets, the rst and nowadays almost legendary one being that of Goldberg et al. [107],
who analysed the scattering of a Gaussian wave packet from potential barriers and wells
as early as 1967. To obtain the wave functions they did not formulate the wave integrals,
but instead solved the Schrodinger equation numerically. The method they used is the wellknown
Crank-Nicholson algorithm that essentially is based on an equidistant discretisation
of the di erential equation. This in turn leads to a set of recursive equations, and to get
initial values for the recursion, the entire system is placed in a box with in nitely high walls.
Clearly the walls must be su ciently far away from the region of interest to avoid re ections
during the considered time span. In addition, the resolution must be chosen ne enough to
follow the oscillations of the wave function.
The Crank-Nicholson scheme has become the method of choice for many other authors, too.
Collins et al. [108, 83] did not shoot movies with it, but used it to simulate the evolution of
a Gaussian wave packet in order to compare their review of tunnelling time de nitions with
numerical results. They reported that the discretisation of the discontinuous square barrier
potential pro le was crucial for the success of the method and had a signi cant in uence on the
results. In the recent past, computers have become powerful enough that this method can even
be employed to generate on-line animations of wave motion, like in the Mathematica-based
example of Robb [109], who combined the convenience of a modern mathematical software
environment with the computational power of a `workhorse' written in C. A software package
solely dedicated to quantum physics is the program of Hiller et al. [110], which besides many
other functions to explore quantum mechanics, also contains a section where the motion of a
wave packet through a given potential structure can be observed.
Somehow di erent to the aforementioned examples is that by Merrill [111], who in the early
days of computers presented a simple BASIC program to study the propagation of waves in a
dispersive medium. Intended | like all other animation approaches | primarily for students,
it required the initial wave form and the dispersion relation to be supplied by the user. By
use of the trapezoidal rule, it then computed the spectrum of the initial wave form as well as
the inverse Fourier transform of the wave at some later time.
Trying to present moving pictures in a written work is a practically futile attempt due to
the obvious lack of motion. There is only one conceivable situation where something like a
contiuously owing time enters an otherwise static book | when a reader quickly thumbs
through the pages without taking the time to read through them. While clearly being undesirable
from the author's point of view, this opens a subtle possibility to still convey some
information, even information a thorough reader would hardly ever recognise. Hence there
are two `thumb movies' included in the outer top corners of the pages. The sequence on the
odd pages shows a Gaussian wave packet tunnelling through a square barrier like in section
4.6. The parameters | for the sake of completeness | are n =4,k=6, =0:8, and D =4.
The displayed spatial range is [,40; 40], and time runs from T = 0 to approximately T = 42.
The barrier has been portrayed as a rectangle, however, its height has no physical signi cance
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