(Issued Wed., Dec 4)

(This is NOT Due - it's just a guide for studying chapter 12)

The final will be held on Tues, Dec. 17, 3:30-6:30 in our regular classroom (G-116) The final may cover any material from the entire semester, but will focus on material since the 2nd exam, i.e. Gas. Ch 7-12. (There will most likely be something regarding the Hydrogen atom.)

Required reading for this week: Ch. 12

1) Gas 12-5

2) Gas 12-6

Note: You only need to "write out an expression". You should be able to do the required angular integrals pretty easily, but you needn't bother actually doing the radial integral. (Do try to set it up, with correct factors of 2 and Pi, and limits of integration, etc.) (I don't believe the integral is even doable in closed form!)

3a) Find <r> and <r^2> for an electron in the ground state of hydrogen. (Express your answer in terms of Z and a0, the Bohr radius). Do not assume Gas. Eq. 12-36, but you may use Gas 12-30.

b) Find <x> and <x^2> for an electron in the ground state of hydrogen. (Hint: No new integrals are required! Use , and exploit the symmetries of the ground state)

c) Find <x^2> for the state n=2, l=1, m=1.

(Note: This state is NOT symmetric. x=r sin(theta) cos(phi) should help...)

Finally, to really test your skills and knowledge of this last material, you could give Gas 12-10 a try.

(At the very least, set up the radial equation. Can you convert to unitless variables? What's the "asymptotic" behaviour at small and large r? What would you do next?... This is plenty (you really don't need to go through the problem any further), but if you're intrigued: in order to get Laguerre polynomials to "appear", you need at this point to switch variables (the trick is to define a new variable z=r^2). In the end, you can find the energies (easy), and degeneracies (more tricky) this way.