Post deleted (see homework for Tuesday post)

I put the content of this into HW for Tuesday (expanded version)...

Jesse's eigenstate sketches and related discussion

Here are some simple pictures that capture the essential nature of the eigenstates. Please comment and discuss here.

Eliot's beautiful eigenstate pictures

Here are some pages from Eliot's midterm. He made some nice eigenstate pictures. His work also shows an excellent sense of graphic design with figures of appropriate size, etc. Appreciate the relationship between these sophisticated plots and the Jesse's very simple drawings in the next post, which express the same essential eigenstate symmetries and can be created with less technology.

Homework for Tuesday, Nov 4 and more (expanded version with free bonus problems!)

30. Calculate the optimal size Guassian wave-function for a delta function potential. Graph and discuss the dependence of "a" on alpha.
---extra: Discuss criteria for deciding how good this wave-function is. Establish a criterion (or more than one) and make a value judgement based on that. Can you come up with a quantifiable assessment using your knowledge?

31. Calculate the optimal size Guassian wave-function for the potential V=g x^2. Explain your results. How good do you think this state is? Can you quantify that?

32. Using a state of the form of the gs of the delta function, but with variable size, calculate the optimal variational gs for a potential proportional to absolute value of x.

33. For the 1D 2 delta function molecule, or the hydrogen molecule with one electron (H+), work on calculating one or more of the 6 terms we delineated and discussed today:
Ac, V_atom, Vx, V_other, T_atom and Tx. The ultimate goal is to add them, minimize Etotal, and thereby see what happens regarding the confinement length of the wavefunction (ca), and where moleular binding energy comes from.

29. (not due immediately) Design a problem involving or related to time dependent perturbation theory.

Molecular state: varying "confinement"; RSVP

Create a "molecular state" for one electron in a 1D potential consisting of 2 attractive delta functions of equal strength separated by a distance d. (This is a very simple model molecule; sort of H+ in 1D.)

Suppose your state is a linear combination of atomic states. Imagine fixed alpha, and that the value of the atomic wavefunction parameter corresponding to alpha is a. For specificity, let's consider two cases: i) where d=a and ii) d=2a.

Suppose your state is expressed as as ...exp{-(x-d)/(c a)} + exp{-(x+d)/(ca)}, where c is a confinement related parameter which can be varied. (While keeping d and a fixed.)

Here is the question for you: There are different possibilities regarding what value of c will minimize the total energy of the molecular state. Define the question by delineating the possibilities, and discuss what value of c you think will minimize the energy. You do not have to be right. Just try to argue/discuss energy in a meaningful way. In fact, you can, and are encouraged to, argue for different viewpoints (in different paragraphs).
(You can hand this in on Thursday, if you like.)

Scientific Discovery and Understanding: pages from McClintock biography

These are pages from a book by Fox-Keller. These pages touch on her method of approaching difficult problems. The 2nd page may be the more relevant.