Prof. Axon's full lecture notes are available in PDF format.
The main ideas for today are
We'll also look at the homework assigned for next week.
Choice of coordinates
Recall that we can apply Schroedinger's equation to a particle trapped in an 3-D infinite potential well:
The choice of an (x, y, z) coordinate system makes sense here: the potential itself fits naturally into the Cartesian coordinates.
But what about the hydrogen atom? It is simply an electron and a proton -- no walls.
Spherical coordinates
Using Schroedinger's equation in spherical coordinates.
As before, we GUESS that there may be a solution which is separable in the 3 coordinates.
The portions of Schroedinger's equation which deal with the two angular coordinates aren't too hard. The solutions turn out to be combinations of sines and cosines, in a form which pops up in many areas of physics and mathematics. These forms are sometimes called spherical harmonics.
The radial wave function
The solution for the ground state is not so bad:
For states with higher energy -- and with non-zero values of angular momentum, so that l > 0, the solutions are more complicated.
Now, for states with no angular momentum, l = 0 and m = 0, the complete wave function is
This purely radial wavefunction yields an energy of the electron which should look familiar.
The resulting energy spectrum is identical to that of the Bohr model.
If we look at the wave function itself, we find it to be perfectly symmetric: the electron is equally likely to be found at any angle.
However, as soon as we introduce some angular momentum, we break the spatial symmetry of the wave function. Compare these views of the wave function for n = 2, l = 0, as seen from two viewpoints:
with this view of the wave function for n = 2, l = 1, seen from the same two viewpoints.
This broken symmetry may manifest itself as a change in the energy level under certain circumstances. For example, if we impose an external magnetic field on the atom, then the energy of levels with different amounts of angular momentum parallel to the magnetic field (different quantum numbers m) will split:
We call this the Zeeman effect.
Angular momentum.
Let's look at the quantum numbers l and m, which correspond to the angular portions of the wave function. They are associated with the angular momentum of the electron's orbit.
