For Thursday Oct 16 (the day after tomorrow...) how about if you turn problem 18, 20 and the -eEx related problem. I'll just outline them briefly here, and if someone can describe them in more detail or from a different perspective, that would be most helpful and appreciated:
22. the -eEx problem: For this problem, since we have already gotten pretty far, due to the wonderful work of today's "presenters" and their friends, I would suggest focussing on the nature of the eigenvectors of the matrix:
0 0 1 1
0 0 0 0
1 0 0 0
1 0 0 0
and what they "tell us" in the context of this problem. Pictures would be important here. Would someone would post something about the basis for this matrix... and anything else relevant...
18. Design your own degenerate perturbation theory problem.
21. For a 2-D H.O., express the angular momentum in terms of raising and lowering operators. Get it as simple as you can. (Start from the definition of angular momentum.) What is the dimension of L in this case. Do the x and y raising and lowering operators commute?
23. (extra credit) Calculate the matrix of L in the basis of the lowest 3 or 6 states of the 2DHO.
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25 comments:
I guess maybe 22, 20 and 23 should perhaps have a higher priority than 18? What do you think?
Are there any problems we have forgotten?
Zack, im sorry to be the one who is asking this, sahil told be too, but what if we all start working on the problems tomorrow. and then thursday you can add another problem or something, and we all turn it in tuesday. The reason is that half the class is taking the GRE saturday, and also, I know most people have not started 18 because they were working on the problems due today. Reading through the problems , it doesnt seem that bad, but the less stress the better. I personally will only be able to work on these problems tomorrow night, and i just dont feel that I will get everything out of the problems if thats the case.. I dont know,.. We are all very motivated and enjoy the hw! we just need some more time maybe.. But maybe its not too much i dont know.
for the Hydrogen matrix, what are the coefficients, ie energy terms etc.
also which states do the eigenvectors {0, 0, 1, 0} and {0, 0, 0, 1} correspond to?
they correspond to psi(200)psi(211) and psi(200)psi(21-1) respectively.
I agree with Megan regarding the homework due date issue. I'm also taking the GRE and would like to spend more time to really make sure that I understand the homework and let the concepts simmer for a bit instead of just racing through them.
Megan and Chaos have valid points. This is short notice for a homework set. I know that I'll be scrambling to get this together and I won't have much time to break the problems down and fully understand what they are saying.
Also, don't we already have a problem 20 that is different that the one listed here?
I don't think I agree with Megan's conclusion that the eigenvectors [0,0,1,0] and [0,0,0,1] are psi(200)*psi(21+1) and psi(200)*psi(21-1). It seems to me that the eigenvectors should tell you what the preferred basis of excited states are for the perturbation such that the correction matrix (terminology?) is diagonalized. I think they correspond to psi(21+1) and psi(21-1). So that a given eigenvector tells you the weighting of the states like this:
[ {psi(200)}, {psi(210)}, {psi(21+1)}, psi(21-1)} ]
The eigenvectors should not be a mixing of states already but rather a recipe to construct a better basis. I might be completely wrong on this; does anyone agree?
I agree with you Bobby. I just wasn't sure which was psi_2,1,1 and which was psi_2,1,-1.
as for homework, I agree with the above opinions. The problem #20 listed here is the same as shown in previous posts as I see/remember it. I also agree with Bobby on the -eEx problem, you are evaluating your original choice of wavefunctions to find an appropriate combination under the (new) correct basis. In response to trapezoidal. The states corresponding to those eigenvectors depend on the way the perturbation matrix was set up, but it seems like they would correspond to Psi(211) and Psi(21-1) respectively, although I wasn't sure there were any resulting stating that were NOT a combination of these states, and just some form of the individual states alone...maybe I'm missing the question. I believe the energy corrections (ie eigenvalues) are 0, 0, and +/- 3eaEPi. Hope this helps...
i think i thought jesse was talking about the matrix elements but i guess he wasnt.
Jesse, just to be clear those aren't the correct eigenvalues for this problem, you are just trying to understand the ordering?
I agree that turning in problems tomorrow is a little much with the GRE saturday. I also agree that #18 should not be the lowest priority. We should just have a more substantial set of problems due Tuesday.
For us worrying about the GRE, stressing about technically finishing these problems asap isn't very productive in terms of learning the material. It's better to have a longer time to think about them, especially #18, after the stress of this exam has been lifted.
Also, for anybody still confused, #20 is #21 from the previous post. I'm going to label mine as #21.
Bobby's right in the response to Jesse's question, Jesse was asking about the original basis wave functions (not the change of basis resulting from the perturbation. Psi(21+1), Psi(21-1) are what he had in mind.
As part of the Hydrogen group, i also confirm that the original basis that we were using was {Psi(200),Psi(210),Psi(211),Psi(21-1)}. Regarding graphing these linear combinations, is the idea to figure out what the different states look like alone and there merely superimpose them (taking into account what their relative sizes should be)? If that's the case, i don't see how combining the states will take care of the imaginary parts of Psi(21+/-1). graphing |Psi|^2, on the other hand would resolve this issue, and is how we normally visualize hydrogen states
With regard to Nina's point: "i don't see how combining the states will take care of the imaginary parts of Psi(21+/-1, ... "
I think this is a very critical point and is worth really thinking about.
I see that there is a lot of resistance to handing in HW on Thursday. Here is a proposal: how about if everyone spends a few hours working on the problems so that we are really prepared to discuss the states that emerge from the -eEx problem and perhaps the angular momentum of the 2DHO, and we won't hand in any HW on Thursday. Does that seem OK?
Also, if you get a chance to create the matrix for -eEz (an E-field in the z direction), that might give you insight into the -eEx problem eigenstates.
Jon, good point about the problem number. I changed it to 21 in the post to be consistent with the previous time...
for the -eEx problem, the eigenvalue of the matrix is degenerate with lambda = 0 (i think it's degenerate because lambda is 4th order and there are only three eigen values, 0, sqrt2 and -sqrt2). for lambda = 0, the second element of the eigen vector seems like it can be any value. Does anybody know why?
Further more, is this telling me that i can weigh the second wavefuntion anyway i like in the combination (with the proper normalization factor)and it would still satify the set of linear equations?
Well Chao, that is a very interesting question. I believe that maybe its because the eigenvalues are degenerate that it results in this manner or maybe it might be the fact that there are a lot of 0's in matrix which results in being able to exploit the freedom of choosing your second element of your eigenvector. However I noticed that there is also a discrepancy with being able to choose any value for your second element, which is that the normalization factor (which you mentioned) will change depending on the second element's value and weight each element of this eigenvector differently due to the arbitrary choice of your second element. I just find this strange because when you choose the values for each component of your eigenvector, the normalization factor should change accordingly to accomodate for the relative size of each component, but this seems different from the case we have here. Well I don't know how to answer your other questions but this is the best possible answer that I can give.
Nina, im confused, i thought that making a linear combination of psi(21-1) and psi(211) would cause the new wave function to be all imaginary or all real, like we did in one of our earlier problems? If something is all imaginary isnt it still considered real? Could we just simplify our linear combinations into sines and cosines and then graph? or would there still be the imaginary part problem with this?
Megan, if you look carefully at the eigenvectors we calculated today for this problem you'll see that Psi(21+1) and Psi(21-1) always combine in such a way to make those cases of being fully real or fully imaginary (which can be considered real). Even though the matrix is calculated with the original real and imaginary states are used it cancels out in the end.
Zack, this sounds like a great alternative to me. I got 18 and 22 done but 21 completely stumped me and I'd love to talk about it during lecture. I tried some method of using the definition of L and substituting in the ladder operators for x, y, px, and py but this didn't seem correct or clear.
Chaos, when you solve for eigenvectors you're solving essentially for a new set of axes along which to orient your wavefunctions. You're correct that for the second value of the second eigenvector you can choose any value but this will be normalized to 1 so that your new axes are unit vectors along the preferred directions. Does that sound right or make sense?
Nina, in combining Psi(21+1) and Psi(21-1) in equal amounts by either addition or subtraction you shift the wavefunctions that exist in partly the real axes and partly the imaginary axes. By adding them the imaginary parts cancel out and the combination is purely real. When one subtracts them, one is left with a purely imaginary psi that can be multiplied by i to give a real psi. This was detailed in problem 9a. Is this what you were asking about?
I think the (unnormalized) probability density of the (Psi_211-Psi_21-1) state looks like this.
Sorry about that, guys. I just got tripped up for a second. I totally see it now. Tim, that looks really great/interesting!
haha. Sorry about that, my housemate hacked my computer. Linnea = Nina
Bobby, I think you are totally on the right track in terms of writing x and p in terms of raising and lowering operators (and the same for y). That is the place to start: express x and px in terms of a+ and a-...
Regarding Chaos' point, i think there may be more to it than just normalization. (Remember, this is the person who pointed out our erroneous track in the last class.) Perhaps it is related to remaining degeneracies, though i have been mistaken before.
In addition to HW discussed above, here are some cool things someone might want to try:
Calculating the matrix for an -eEy perturbation for the H-atom 1st ex. states.
Same for -eEz
diagonalizing Lz for the 2D (or 3D) H.O. for the 1st excited states or 2nd excited states.
What is the effect of a magnetic field on a 2D HO? Does it lift degeneracies? Does it "break a symmetry"?
What is the effect of delta funtions on each axis (at x=+-a and y=+-a) on the degeneracies of the 2D HO? (picture 4 delta functions, all equivalent, so you* still have invariance for 90 degree rotations but not rotational symmetry (for arbitrary rotations). ("four-fold" rotation symmetry around the z axis. Does that make sense.)
*by you, of course, I mean the Hamiltonian, haha
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