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The hydrogen atom is the simplest possible atom. It is made of a proton with positive electric charge and an electron with negative electric charge. The two are attracted together via the electric force. We can represent the force on the electron exerted by the proton using our good friend the potential energy diagram introduced during the "light as particle" week.

The electric potential energy felt by an electron around a single proton is just a "well" that gets steeper the closer the electron gets to the proton. Using r for the radial distance the potential energy looks like the Figure "Potential Energy for Electron in Hydrogen"

Notice that we have defined the potential energy U to be always negative. This is an arbitrary choice. It is also a useful choice although it may seem confusing. Here is why we do it: The total energy is the sum of potential plus kinetic, so E_{tot} = U +K . If the total energy of the electron is a negative number, it will not be possible to escape from the atom since there is a turning point in the trajectory. We say that the electron is bound. If the total energy is positive, then the electron can escape to infinity and it is thus free of the proton. This happens when we ionize hydrogen and liberate the single electron. Ionize just means "turn into an ion" by freeing an electron. This is illustrated in the "Negative Energy" figure.

Quantization and the Binding Energy of Hydrogen

The electron is bound to the atom just like the electron was bound to be in the box that we discussed last week. The shape of the box is different now but the basic idea is the same.

• The electron is confined, it cannot be on the proton and it cannot be further than its turning point (which depends on its energy and the shape of the energy graph for the atom).
• Because of these limits, only quantized standing wave patterns are allowed.
• This means that the energy of the electron is quantized. Quantized means only specific energies are allowed, the rest are not.

Physicists can solve exactly for the standing wave solutions. Because the solutions are in three dimensions, it is complicated to write a formula or even just to visualize them. Below you can see standing waves on a drum (this is two-dimensional).

Instead of showing you the wave, we will show you the square of the amplitude of the wave (see Fig "Electron Wavefunction for Hydrogen"). I remind you again that this picture shows you where the electron is most likely to be found, not where it is.

Because, we are in three dimensions, there are three quantum numbers (three integers), n, l and m. In this class, we will simplify and only keep track of n, which we call the level.

External Resources.

• The First Image Ever of a Hydrogen Atom's Orbital Structure An image of the orbital of the hydrogen atom!