As electrons move, they interact with other objects. For example, an electron in an atom interacts with the nucleus. In an electron microscope, the electrons interact with magnetic fields as they move toward the sample. As we develop wave functions, we must include these interactions as we learn about representing electrons with wave functions.
Because Schrödinger's Equation is based on energy, we will represent interactions with potential energies. Locations where the potential energy changes indicates places where the interaction changes. A potential energy of zero indicates that the electron is not interacting with anything. Positive values of potential energy will be used for interactions in which the electron is repelled by other objects, while negative values will be used for interactions in which the electrons are attracted by other objects. To represent potential energies, we will use graphs of the potential energy versus location -- a potential energy diagram. One such diagram is shown in Figure 2 below.
Consider a beam of electrons that are moving in space, for instance the electrons in a TV set. From the back of the TV picture tube the electrons travel in a vacuum until they reach the screen. In that region they do not interact with other objects. When they reach the screen, they encounter a large number of other electrons.
The repulsion between the electrons in the beam and those in the screen cause an increase in the electrons potential energy (Figure 2). In our example we have shown an increase of 2 eV for the potential energy. The total energy of the electron (3eV), however, does not change. The total energy is represented by the dashed horizontal line in the diagram, the potential energy is represented by the solid line.
Figure 2: Potential & Total Energy of an electron approaching a sample.
What is the Kinetic Energy of the electron in the vacuum in Figure 1?
in the metal?
Calculate its de Broglie wavelength in each region.
We covered de Broglie's wavelength in Module C, Tutorial 3. If you can't remember the equation it is shown below:
Since h = 4.14 ´ 10-15 eV and mass of an electron is 9.1 ´ 10-34 kg, for electrons only we can write:
As you can see from your calculations above, the de Broglie wavelength changes when the potential energy changes. Thus, we will divide the space into regions based on the changes in potential energy. Any location at which the potential energy changes is a boundary between regions.