May 14 class & additional notes.

Tuesday's class is important. We will cover H atom states and things you need to know
to tackle HW 7. The electron states of the H atom system are highly unusual. They are many ways to view and utilize them.
1. Notes on energy of single electron states in an H atom potential:
The ground state is an energy eigenstate with an energy of about -13.6 eV.
1st excited states have an energy of about -13.6/4= -3.4 eV.
There are an infinite number of these bound-state energy levels. Their energies are given by \(E_n = -13.6/n^2 \: eV\).
There is only one ground state; the space of 1st excited states, belonging to the energy eigenvalue \(E_2\), requires 4 different 1st excited states to span it. 1+ 3 + 5=9 states are needed to span the space of 2nd excited states belonging to the energy eigenvalue \(E_3\). We say that the degeneracy of the energy eigenvalue \(E_3\) is 9, and the degeneracy of the energy eigenvalue \(E_2\) is 4.
     To sum up, there are a lot of linearly independent (orthonormal) bound states (a countably infinite number) each with an energy less than zero. The ones of primary interest are the lower energy ones, particularly those with n = 1, 2, 3 & 4.
     Each of these states is a single-electron state. We are talking about states of one electron. Even when we add states together to create a superposition state (mixed state), that is still a one-electron state. We have not introduced the way to make states of more than one electron yet. When we do, you will definitely notice because it will be different from what we are doing now and more complicated.

2. Notes from Tuesday. There are 3 pages of notes and 2 wolfram alpha plots. The wolfram alpha plots are just to encourage you to use that yourself to examine and explore the nature of wavefunctions and \(\psi^*\psi\).  Note that the contour plot is for a mixed state involving a superposition of the ground state (1s) and the 2x state. That is a time-dependent mixed state. The contour plot for that is at t=0.
    The "path" plot, on the other hand, is for the 2s + 2x state which can be viewed as a superposition, given a prior choice of basis, but is also an energy eigenstate.

 Please feel free to point out mistakes, and to comment and ask questions as you wish.