Problems for summer, atoms, screening...

Here are some possibly interesting problems that we may not have time to fit in this course.
1. Calculate the expectation values of r and 1/r for an electron in the ground state. This involves two different calculations and I think one is not necessarily just the inverse of the other.

2. Do the same thing for the 1st excited states. Just 2x and 2s should be fine, since 2y and 2z will be the same as 2x. From 1/r you can get the expectation value of the PE.
a) Is the PE for all four 1st-excited states the same?
b) Is \(\langle r \rangle\) different for an electron in 2s and 2px? (I think that it is different and that that difference may be the opposite of what you might expect or guess. That is, I believe, though I could be wrong, that \(\langle r \rangle\) is larger for the 2s state.)
In discussing atoms with more than one electron people sometimes talk about screening? What is screening? How is it related to the Schrodinger equation? (It is related, but through what term?)

3. Calculate the expectation value of the kinetic energy for the gs, 2s and 2x states.

4. Show that an electron in the mixed state: \(\Psi = \frac{1}{\sqrt{2}} (\psi_{1s} + \frac{1}{\sqrt{2}}(\psi_{2x} + i \psi_{2y})\) has a circular trajectory. What is the period of the orbit? What is \(\langle r \rangle\) in nm? What direction does it rotate? How would you make a state rotating the opposite way? Is that state orthogonal to the other one?