As we discussed today, the periodic table originated from considerations of bonding, with elements arranged under elements with similar bonding behavior. With the development of quantum mechanics, and, specifically, considerations associated with the degeneracy and angular momentum patterns of one-electron states which are solutions to the Schrodinger equation for hydrogenic atoms (-1/r potential), there is a theoretical basis for understanding the periodic table.
In your assignment, you are asked to explore and establish the nature of the periodic table for harmonic oscillator potentials in 2 dimensions (and 3 dimensions as well if you can). Create a periodic table which might be appropriate for a HO type potential (r^2). Compare and contrast that to the periodic table for -1/r.
Aside from the central issue of the difference in the nature of the periodic table (if they are different), one could also address related intriguing issues such as: How do the degeneracy patterns different? Where do these degeneracies come from? (Why are states of different angular momentum sometimes degenerate?)...
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How is it going with this? If anyone is stuck or confused they can email me. (zacksc@gmail.com)
There is one HW problem that perhaps not everyone finished that might be relevant, especially if you doing the 3-dimensional harmonic oscillator. That is the one where one looks at the effect of (6) equal delta-functions (each on an x, y, z axis (+-)). I was thinking that that problem would lead you to the discovery that the H-atom l=2 states (d states) can be combined to make states proportional to (x^2 - y^2), xy, yz. zx, and one more (something like 2z^2 -x^2-y^2 ). Knowing about that relationship might help with intuitions regarding the H.O. problems.
Alternatively, one can use powerful mathematics to analyze everything, but the matrices do tend to get large fast for the 3D case, so intuition could be helpful.
Recall that for the 2D H.O., the eig-functions for the 2nd exc states, as constructed from the 1D orbitals, were (proportional to): x^2, y^2 and xy ; and these could be combined to make x^2-y^2, which is a rotation (45 deg) of xy, and (x^2 +y^2) . The L=2 states could be made from out-of-phase combinations of x^2-y^2 and xy [ 1+i, and 1-i], and, of course the x^2+y^2 is independent of angle.
Do we consider this as a normal homework? I'm a little bit confused.
I think we decided, in class, to elevate it to the status of "take-home midterm". Essentially, I guess that means that you work on this gets a particularly close reading from the "professor" and that it has an enhanced role in helping us see how well you are understanding things. Does that make sense?
(So, like, do a good job on it.)
Thanks for the helpful comments, Zack. Since this problem has elevated status, should we turn it in next tuesday since thursday is a holiday? and the 3D case is optional?
i think the plan was to turn it in tomorrow. (We only have 3 classes left, including tomorrow.)
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