In our descriptions of electrons so far, we have been assuming that electrons move in the beam and all of them have identical energies. This situation has been useful, but it is somewhat artificial. For a real world we need to describe individual electrons as well as beams.
First, let's see why our present version has problems when dealing with an individual electron. Suppose we wish to use a wave function that travels straight across this line of the page. It moves from left to right without changing its speeds. It interacts with nothing.
So, the potential energy is zero. One electron, no interactions, and no energy changes. Sounds like a simple situation.
The Wave Function Sketcher program would give us the result in Figure 1.
Does this wave function match the description above? Why or why not? (If you have difficulty with this question, think about the probability of finding the electron at various locations.)
The difficulty is that the probability density is rather uniform across the page. Also, it does not change with time. Let's create a better wave function for a single electron. Suppose you were to sketch an approximate graph of probability density versus position for an electron that was say, at approximately position 300nm. The approximate shape of the wave function, not the numerical value at any point, is the important feature.
Describe the sketch you made below.
Explain why you drew the wave function as you did both near the location of the electron, and far from it.
How would you expect this wave function to change as the electron moved to the right?
A possible representation for the electron moving across the screen is shown in Figure 2.
The program Quantum Motion shows an animation of this.
Instructions: 1) Set the potential Height to zero (we now have a free electron). 2) Click the triangle option and then click the square option 3) Press the Play button.
This wave function has a probability different from zero only in a small region. It moves as the electron moves. So, it fits the need for our purposes.
All measurements indicate that the electron is a very small object. It is essentially a point in space. So, the "bumps" on the wave functions in Figure 2 indicate that the probability of finding the electron is different from zero at several different locations.