Schrödinger’s Equation & Potential Energy

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 situation is shown in the diagram.

 

To see how these diagrams help us with the wave description of matter, answer the following questions:

 

What is the Kinetic Energy of the electron in the vacuum in the diagram?

 

What is the Kinetic Energy of the electron in the screen?

 

Calculate its de Broglie wavelength in each region.

 

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.