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.
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11 comments:
Has anyone done 31 yet? I'm getting an imaginary answer for the width which doesn't make any sense. I've run through the calculation twice now and haven't found my error. The kinetic energy might be wrong since that integral is so involved. I've got -3*h-bar/(4*m*(ca)^2) for my T. Is anyone else getting something different or the same?
A follow up: I'm actually getting a complex answer because i^(1/4) is (1/sqrt(2))*(1 + i). Do I just throw away the imaginary term? If I were looking at a time-dep case would I treat this as oscillatory?
Can somebody help Bobby with 31.
regarding 33, I think it is less difficult than I though and everyone should be able to do all the integrals without much difficulty/ Then the key thing is to try and look at what they are telling us. I would avoid the region d less than "a" for two reasons: mainly, our wavefunction is of a form which makes more sense when the two potentials are not too close together. Also, we have no analogue of the proton-proton repulsion that one would have for a real molecule. Unless someone can come up with somehting that would work?
Anyway you can calculate each term in "closed form" and the graphing and/or "tabling" them and looking at their values, as a function of d for fixed c, or as a function of c for fixed d, or ... may be interesting. (Maybe try using excel.)
The most important part is to look at the results and interpret what they are telling us!
From other post:
"...we can explore some plots of all 5 of the terms in E_total (individually and added together). I think the plot range should be something like d going from a to 10a; and I would suggest trying that for c=.9, 1.0 and 1.1 for starters (to test our hypothesis...). Does that make sense?
Another way to start might be to fix d, at about 3a or so maybe, and then plot the individual terms and E total vs c....
Please post results here as you get them and we can discuss them.
Questions and comments welcome. "
Bobby I got (h_bar^2/2m)*(1/ca)^2 for my value of < T >
when graphing a vs. alpha should we be fixing the value of "c"?
Going off what trapezoidal said, should it be "ca" that we want to compare to alpha or else I don't really know how to isolate 'a' by itself without 'c'.
Ok, thanks Jesse(? who is trapezoidal again?). I'll look through my calculations again.
When it came to graphing I graphed E vs ca. It doesn't make sense to try to minimize just c or just a since they both tell us the width.
I think the idea was that with "c" as the variable parameter, then we could have a fixed "a", and that way we would notice if the length scale, ca, got bigger or smaller than the "atomic length scale", a (c=1). It is just a notational convention. (We could have just let "a" be a variable parameter instead. Then we wouldn't need c.)
Regarding 33, the 1D molecule problem, it seems like the integral of psi(x+d/2) psi(x-d/2) comes up in a couple of places, i.e., in the calculation of Ac and of Tx. It sort of naturally divides into 3 regions, below -d/2, -d/2 to d/2, and above d/2, and i think you end up with:
(1 + d/ca) exp[-d/ca]
Does that seem right? Did anyone else get that?
With that, I think Tx is negative for all d/ca . Did anyone else get that?
*btw: That integral appears in Tx because the 2nd derivative of psi is, i think, psi/(ca)^2 + "the delta-function term" (which is negative). Does that seem correct?
I hope that things are going well with these problems. 30-32 are all problems in which the opposing propensities of T and V are manifest, and from their competition one finds an "optimal" state. I thought of an interesting enhancement for 32. For that potential, which would make a better variational state? Something of the form of the: 1) the delta-function g.s., or 2) the H.O. ground state?
What does your intuition say about that? How would you define "better"? How could you answer this question?
33 is interesting in that it is a "molecule problem" for which the calculations are not very difficult. For a model H2+ problem, where you construct a similar state from H atom states centered at different sites, the form of the problem is identical (i.e., there is a Vx, Tx, and all the other terms), however,the integrals are much much more difficult (and 3-dimensional). [If a group or individual wants to do that in the next couple weeks that would be welcome.]
Anyway, i am not sure how interesting 33 will turn out to be, but it is a problem everyone can do. If you would like a specific suggestion of how to approach it, I would suggest first doing the integrals and then evaluating each term at a particular fixed value of d -- Something like 2a, 1.5a, 2.5a or...? For that d value, examine the 5 terms of the energy for a few values of c, say 0.9, 1.0, 1.1, and then think about comparing the energies you get to that of an isolated delta function with an electron. It is in that comparison that the molecular binding energy emerges, i think...
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