Introducing Wave Functions

So far we have seen two results of the double slit experiment with electrons

  1. Electrons behave as waves causing a pattern of light and dark regions to develop on the screen.
  2. It is not possible to predict where any one electron may appear on the screen when it is sent through a double slit. However, we can state the probability of it appearing in various locations.

Now we will learn how results 1) and 2) are related to each other and develop a single concept that is consistent with both results - Wave Functions.

Because matter waves are abstract ideas used to describe results, they do not travel through a medium, such as water. In fact, a matter wave is not a physical entity at all. So, scientists generally describe these waves in terms of mathematics or pictures. (We will use the pictures.)

Physicists call the pictures or equations wave functions. A sample of a wave function of an electron is shown below in figure 2. The shape of the wave function is NOT the path that the electron traverses. It is a useful description related to the location of an electron and can be used to predict the probability of finding the electron in any given region of space.

Figure 2: Example of a wave function for an electron.

A large amplitude is related to a large probability while a small amplitude goes with a small probability. We have a slight complication. Waves can have amplitudes that are either positive or negative; probabilities must be positive numbers. The solution is to square the wave function. A square number is always positive. The wave function squared appears below in Figure 3.

Figure 3: Probability density graph from the wave function in Figure 2.