You have completed the steps in creating the wave function. The next step is to interpret it in terms of probability of the electron at various locations.
For your wave function of the electron encountering a metal, sketch the probability density below and write a short paragraph about what you can learn from it.
We have now established a way to create a wave function when we know the physical situation. The four steps require that we set up energy diagrams; determine the electron wavelength, adjust the wave function to be smooth across the boundary and interpret the results. From this process we can predict probabilities concerning where electrons will be. We will continue these procedures in the next activity. We will consider a different situation and come closer to understanding why atoms can have only certain energies.
In summary the four steps are:
Step 1. Consider the physical situation, draw the energy graph and determine the boundaries. Step 2. Determine the wavelength (if possible) in each region and sketch the wave function. Step 3. Adjust the phase and amplitude of each part of the wave function until it is smooth across the boundary. Step 4. Interpret the wave function and discuss the probability of finding the electron at various locations.