We could continue this process of stepping through the energies and eventually find all acceptable wave functions and energies. However, the computer can be made smarter. It can complete the process automatically. It can quickly try hundreds of wave functions, reject all that have high probability densities outside the atom and keep the rest.
Select Energies/Search Allowed Energies from the pull-down menu. Horizontal lines corresponding to each of the allowed energies appear.
Record the value for all allowed energies as well as the potential energy depth and width of the atom.
The energies which are shown on the screen are obtained by solving Schrödinger's Equation for the potential energy on the screen. In obtaining these energies we have used
With these assumptions we see that only certain well-specified energy allowed for the electrons.
From Module B we know that changes from one energy to another causes light to be emitted. From the results here we see why only certain energies are allowed. The wave nature of the electron requires it.
Our present potential energy model is somewhat simplified. Thus, we do not get the correct values for the energies. Really doing it right would require three dimensions and a more accurate potential energy. However, the basic conclusions would not change; only the number would change.
You can view the wave function corresponding to any allowed energy. Click on the total energy (the horizontal dotted line) outside the vertical potential energy lines. The wave functions corresponding to the energy level appears to the top right. The program shows both wave functions (symmetric and antisymmetric). Only one is good. You must decide which one.
Hint: Remember the wave function must be smooth and decay to zero in classically forbidden regions (where Potential Energy is greater than Kinetic Energy).
Sketch the acceptable wave function for the lowest energy level, describe it below.
"Keep" the wave function. Now view the wave functions for the highest energy.
Sketch the antisymmetric wave function for the highest energy level. You should notice that this is the acceptable wave function for this energy
How are the wave functions of these two allowed energies that you have sketched above similar to each other? How are they different?
Problem
Using the width of the potential well and deBroglie's wavelength equation, calculate the Kinetic Energy of the electron in the lowest and highest energy state.
HINTS 1) deBroglie's Equation is:
Wavelength = Ö(Planck's constant)2/ (2 ´ mass ´ Kinetic Energy)
Since h = 4.14 × 10-15 eV and mass of an electron is 9.1 × 10-34 kg, for electrons only we can write:
Wavelength = Ö1.5 nm2/ Kinetic Energy
2) You should be able to see a relationship between the wavelength and the potential width.
In the box below describe how you calculated the kinetic energy for each energy and give your numerical values. If you feel totally confused by this question write a note to your instructor in the box.