We further will investigate wave functions of electrons in atoms. By looking at several possibilities we will see why different atoms have different spectra.
Using the program Wave Function Sketcher to find the "special" or "discrete" energies for atoms will be rather tedious. So instead we will switch to another computer program. The program follows a procedure similar to the one we used with Wave Function Sketcher. It is faster than we are and does not get tired of doing the same thing many times.
Start the Bound States program (Figure 4).
It starts with a potential energy similar to our model. Select a total energy for the electron by clicking anywhere on the potential energy diagram on the left side of the screen. A red line appears on the potential energy diagram that indicates the value of the total energy.
For any energy, the program can determine two possible wave functions. One of these wave functions has a zero in the middle of the negative potential energy region, while the other has a maximum there.
Both wave functions are solutions for the Schrödinger equation. Physicists label these equations as symmetric and antisymmetric. Those terms have little meaning for our purpose, but we have to give them a name. So we use symmetric (maximum in the middle) and antisymmetric (zero in the middle).
In this mode Bound States is both smart and dumb. It is smart because it can solve Schrödinger's Equation. For any energy it creates a wave function with smooth connections at each boundary. It is dumb because it pays no attention to physics. The wave function may not make sense.
To think about the physics consider the location of an electron in an atom. Is that electron likely to be near the nucleus or very far away?
Now, look at the wave function in Figure 5.
We have another way of rejecting this wave function. It does not fit our steps. Its value does not decrease to zero when the potential energy is greater than the total energy. Both ways say the same thing. Common sense tells us that an electron should probably be near the nucleus. Wave functions that say otherwise are no good.
So, the programs can be dumb. But, it can help us find acceptable wave functions. Use the up (­) and down (&hibar;) arrow keys on your keyboard to slowly change the energy level. The energy level will change by the Energy Step, (= 0.1 eV) each time you press an arrow key. To increase or decrease the Energy Step by, use the left (¬) or right (®) arrow key on your keyboard change the Energy Step to 1.00eV or 0.01eV. Change the energy level until you notice that the at least one of the wave functions is consistent with the requirement that the wave function goes to zero outside the atom.
Make a sketch of this wave function and describe it below.
Press the Keep button to store the wave function. Continue using the arrow keys and find another acceptable wave function. Make a sketch and describe it below.
How does it differ from the previous acceptable wave function you observed?