The wave function sketching program provides a way for students to sketch on the computer stationary state wave functions for any potential in which they are interested. The program, as it stands now, does not teach students how to sketch wave functions but assumes that they have learned a set of procedures for the creation of a wave function. These procedures are outlined in our multimedia and text documents. Once they know the procedures this program will provide an aid in creating the wave functions.
At first glance one might conclude that freehand sketching would be as useful pedagogically as using a program of this nature. In fact, our first approach to the qualitative sketching of wave functions was strictly a freehand approach and was very similar to one described by Taylor and French in the American Journal of Physics in the early 1970s. However, we found that students would make very small changes in their sketches that would obscure some of the subtle nuances of quantum mechanics. For example, when the students would sketch a wave function in a square well, they would vary the wavelength slightly as they moved across the well. Thus, any energy would be possible because these small variations would allow the boundary conditions to match at both ends of the well. The wave function sketcher restricts that freedom and thus students can actually see how the boundary conditions lead to limitations on the allowable energy states within a potential well.
In using the program the students' first step is to create the potential. This potential can be created from a selection of the default potentials, a freehand sketch, or the potential creation program. Once the potential is created the students are to define the regions for which they wish to work. In general we define the boundary of a region as any place where the potential changes. At this time we have not developed a means to work with continuously changing potentials. The students select the boundaries of the regions by clicking the mouse on the axis and then indicate when they have finished selecting regions by clicking the done button. Now they are prepared to sketch the wave functions. For each region the students must determine whether the wave function is an oscillating wave function or a decreasing wave function. Then, within each region they can change the variables of the wave function until they get one which matches all of the rules of wave function behavior. No judgment is made by the program as to the acceptability of the wave function. It is strictly a tool which the students use to create their own wave functions.
As students are creating wave functions they generally match the boundary conditions at one boundary and then work on the next boundary of the wave function. In many cases they will find that making changes at the second boundary affects the results at the previous one. Thus, the process of finding an acceptable wave function becomes an iterative process to create a wave function which actually is acceptable in their potential. This procedure shows quite clearly why a limitation exists on the wavelengths of wave functions in wells. Students can select wavelengths for which no acceptable boundary conditions can be created at both edges of the well. Then, their only choice is to vary the wavelength. Thus they see quite dramatically and directly that energy must be quantized.