Unfortunately, such a wave function does not just "pop out" as a solution to Schrödinger's Equation. So we must learn to build it from other solutions.
The ability of waves to interfere both constructively and destructively enables us to combine several different waves to create waveforms with useful shapes. Because waves interfere with one another both constructively and destructively at the same time but in different locations, we can never create a wave function that is not zero at one location, but zero everywhere else. So, to represent a single electron, we must construct a wave function that, when squared, gives a probability density with one maximum, decreases sharply away from that maximum, and then is zero at relatively large distances for where we expect the electron.
Even simple waveforms like the disturbance on the rope can be constructed by adding simple waves together. Our goal will be to combine simple waves to produce a wave function that, when squared, gives a smooth probability density graph similar to Figure 3:
To see how we might create such a wave look at the addition of waves shown in Figure 4.
By adding hundreds of waves with carefully selected wavelengths and amplitudes we can come very close to the form in Figure 3.
From the de Broglie relation we know that wavelength is related to momentum. So, when we add wave functions of different wavelengths, we are adding wave functions, which represent objects with different momenta.
To create wave functions like this use the Wave Packet Explorer. This program allows us to add wave functions of different momenta (wavelengths) rather easily.
Click in the upper left window (Amplitude vs. Momentum graph).
A vertical line appears representing a wave function whose amplitude is proportional to the length of the line, and whose momentum is the value at the line on the horizontal axis. The wavelength is calculated from using the de Broglie relation. The wave function appears directly below.
Repeat this procedure by first clicking on the graph in the upper left window, at any value of position and amplitude and creating a new wave function. Each time, a simple wave function is added to all the wave functions that were already present. The individual waves are shown on the graph on the bottom left while the sum of these waves is just above these.
How does the resulting added wave function change as you increase the number of simple wave functions?
Now, instead of adding just a few wave functions, add a large number of them. Click many times in a definite pattern to create a large number of wave functions. Make a sketch of the resulting wave functions. Describe your sketch below.
Make a sketch of the Amplitude/Momentum graph, describe it below.
Click on the Keep button on the right of the screen to save your wave function. Repeat the exercise twice, choosing different amplitude and momentum values. Save your wave functions and make sketches of your graphs.
Compare your three resulting wave functions. Describe how the momentum-amplitude graph relates to the resulting wave function.
Some of your wave function patterns are probably similar to Figure 3.