The following are the two problems of your midterm. The idea is that everyone should work on their own, except that you can ask questions on this blog. Please feel free post comments and questions here.
1) Suppose that your unperturbed Hamiltonian is an isotropic 2D harmonic oscillator, and that your unperturbed eigenstates are chosen to be those made as products of 1D H.O. eigenstates.
Consider a perturbation potential which consists of 4 "repulsive" delta functions of equal strength located along the diagonals of an x-y coordinate system whose origin coincides with the center of symmetry for the original H.O.. Assume that they are all the same distance from the origin and that they are fairly far from it, e.g., 2a, 3a, 4a...something like that.
Calculate the effect this perturbation on the energies and eigenstates of the 2d HO, including the ground state, 1st excited states, and 2nd excited states.
Describe and discuss your results. What is the effect of this perturbation on the symmetry of the Hamiltonian and of the states? In presenting and discussing your results please engage your sense of physics aesthetics and emphasize aspects of the problem that seem most surprising, interesting or important. Please use a combination of text and figures (hand-written and hand-drawn is fine) to present your results in an insightful, clear and cogent manner.
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2) Again, suppose that your unperturbed Hamiltonian is an isotropic 2D harmonic oscillator, and that your unperturbed eigenstates are chosen to be those made as products of 1D H.O. eigenstates.
What is the effect on the energies and eigenstates of a perturbation k*Lz, where Lz is the z-component of the angular momentum and k is a small parameter.
Same comments as above regarding what states to consider and the nature of your discussion...
Regarding weighting of the problems, I am thinking that 1) is about 60% and that 2 is 40 % weighted, based on my feeling that 1) may be bit more challenging and interesting than 2) (though i could be wrong about that).
21 comments:
Dear Zack,
Just to make sure, you want all four delta functions all together as a perturbation, correct?
yes
zack, I want to clarify if the "a" in the first problem is the same as the 'a' as we use for the length scale or does it not have to be related?
It is the same "a". i am saying that it should be somewhat far (not too close) to the origin relative to that characteristic energy scale of the problem.
[regarding the history of that statement in the problem, it is an afterthought and just a side issue, I think, but i realized after creating the problem that there are values of "a" that could lead to results that may be pathological and/or weird (though i haven't really explored them). (Basically i wanted to keep the delta functions away from the any radial nodes...)]
The main thing in the problem has to do with the symmetry of the perturbation and how that effects some states in very interesting ways...
Zack, I was wondering, did you want us to do GS, 1st, and 2nd excited states for *both* questions? The second question wasn't very clear about what states we should do. I was planning to just do the GS and 1st excited but I just now realized that you didn't specify. Would you like us to go to the 2nd excited state for problem 2 as well?
Bobby, Zack does mention in #2 that we should consider the same states & nature of discussion as from #1
-E
.... I don't know how I didn't see that.... I'm going to delete my earlier post. Thank you Eliot.
Don't delete your post. It is a good question.
Also, you might consider doing some sort of comparative analysis of your results (eigenstates) for the two problems, if that seems interesting.
Hopefully this isn't giving anything away, but in my calculations I've encounted imaginary eigenvectors.
I'm stuck on how to graphically discuss these states. Did we talk about this in class or am *I* imagining things? Do I simply treat it as a real quantity? graph |psi|^2?
Treating it as real seems strange to me, and graphing the prob. density would lose information about the states...hmm..grasping at straws I guess lol.
Elliot,
I am in the same predicament. I remember adding the eigenvector to produce a real eigenvector. I think this is the only way to physically analyze it.
Adam
Adam, that was the next thing I thought of as well, but I'm running into a new problem.
My new, imaginary, states are not energy-degenerate. So when adding/subtracting these states we're not getting a new energy eigenstate. hmmm
Maybe what you mean is that you have complex eigenvectors??
In that case I would suggest thinking about each eigenvector very intensely and somehow finding a way to describe what they are telling you. Don't restrict yourself dogmatically to only what we have discussed so far about eigenvectors. That may not be adequate for all cases.
Spend time with the eigenvectors. Get to know everything about them.
This process is hard to describe, but I posted some pages from a biography of the geneticist Barbara McClintock, which try to describe the process of "seeing" things clearly that start out seeming complex or unapproachable. Perhaps this might someday seem helpful or relevant? (See next post; images can't be uploaded into comments.)
I'm getting some complex eigenvalues and eigenvectors too. What I did was graph ψ^2. Unfortunately ψ^2 still had an imaginary part for me; in these cases I plotted just the real part. I figured I could attribute the imaginary component of ψ^2 to some sort of phase factor between the eigenstates. For the last part of problem 2, two of my eigenstates had real parts that were exactly the same but differed in the complex planes. I explained this by imagining one starting out purely imaginary while the other was purely real and the two would rotate through real and imaginary with a 90 degree phase difference. Combining the two to get real states is out of the question because of what Eliot noted.
I'm really struggling understanding what the perturbation for problem 2 means. In problem 1 the perturbation just tells the particle to stay away from these 4 points. In this problem though, we're saying if a particle has angular momentum we give it more or less? Could it be analogous to a charged particle (in a 2d HO) with some sort of magnetic field applied to it? What is everyone's view on this?
Bobby, I think that you can arrange your matrices so that you get complex eigenvectors, but your eigenvalues are all real. getting complex/imaginary eigenvalues implies that your energy corrections will be complex/imaginary (unless of course, you end up multiplying your eigenvalues by something that makes the energy correction real. That seems fine too).
I don't think you were having any problems with your solutions, but i thought i'd post this anyway because it was something i encountered aswell.
I would like to choose not to comment on either of the above, but I do have one thing to add vis-a-vis getting to know your eigenvectors. If you have eigenvectors and you are not sure how to get more intimate with them here is something you can try (on your 1st or 2nd date).
Try combining your eig vec. with the ground state; (i suggest the g.s. because it is the simplest (most "vanilla") state.) Then look at something like expectation value of vector r. For the gs by itself that would be zero, right? So what vector r does, as a function of time, may tell you something about the eigenstate you are interested in.
It is a lot of work, but if you are desperate, try it. (Or just sort of "gedanken" it...?) Good luck.
There are also things you could do associated with projections or representations in terms of another basis. Those are kind of sophisticated, in a way, but could also be simple at the same time.
PS. What do you think of the Barbara McClintock stories? The entire book is on google books...
Hi class,
I have a small language problem: whats the actual name of the 'x' operator? place-operator? space-operator? I only know that 'p' is the momentum operator...
Tim, the x operator is formally referred to as the "position operator."
Zack, I'm a little confused about your tip; combine the *eigenvector* with the *ground-state*? I didn't think this was even possible. If you're combining things shouldn't they match, at least on some sort of type classification? I'd understand combining the new basis state the eigenvector details with the ground state but you're losing me otherwise.
I'm reading the McClintock pages now... I'll tell you what I think in a bit.
I mean the gs for the 2D HO. Is that what you are asking? (Did you think maybe i meant the gs for the 1d HO? You are right that that wouldn't make sense.) You can combine a state with the g.s. of the same system to make a "mixed state". (and then look at the expectation value of the position operaror, momentum operator, or something else.) It is a way of exploring, i guess.
Don't forget to think deeply about the eig states of problem 1. Those are my favorites, i think.
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