So to answer the question, why is it easy to draw a wave function
for certain potential and total energies and not for others?
Our conclusion is: Only certain sets of parameters work to give an acceptable wave function. The potential energy minimum, its width and the electrons total energy cannot be any value. They must have the right numbers.
The potential energy parameters are determined by the physical situation. The force of attraction between electron and nucleus determines both the minimum value and the width. For example, suppose we are considering hydrogen atoms. Then the width and minimum will be set by the electrical charge on an electron and a proton.
We have only one variable --- the total energy --- to change. We select the total energy so that the wavelengths just fit. Then we can create an acceptable wave function.
The wave function of an electron in an atom is restricted. Its wavelength must be just right to fit in that atom. Wavelength is related to energy. Thus, the energy can have only certain values. An electron in an atom has very limited possibilities for energy. This conclusion comes directly from the wave behavior of electrons.
We will now pursue this limitation on energies. Then, we will
have everything we need to explain our observation in Module B:
Atoms emit only certain energies of light.