This includes a reposting of problem 14, which supercedes (replaces) the previous post, as well as problems 15, 16 and 17.
Please keep all your homework, after you get it back, in a sequential Homework portfolio, which you or I will be able to examine at any point in the quarter.
For problem 14 try to think about and address the question of the crossover energy scale (e). This can be meaningfully addressed from examining your equations from part a) without necessarily doing any numerical work. You may not be so sure, without numerical solutions, but you can still offer your best ideas about what the crossover scale is likely to be (in terms of a comparison of two (competing) energy scales.)
30 comments:
Problem 18: Please create (and hopefully solve) a problem which uses degenerate perturbation theory. (I know I do not have to tell you that) this would involve a starting Hamiltonian that has some degeneracy and a perturbation that, hopefully, lifts that degeneracy.
Consider your esthetic in designing your problem. What role do things like: interesting symmetry breaking, a sense of surprise, solve-ability, simplicity, demonstrating something or something else have in your problem?
I didn't know quite where to post this, but in terms of conventional physics, once and for all...for spherical coordinates:
theta goes from 0-2pi and phi goes from 0-pi...Right?
i'm... not so sure about that...
what do other people think?
Regarding HW, we have several new problems involving degenerate perturbation theory, and the "spectroscopy" problem in which we look at the effect of fine structure related adjustments to the H-atom energies on the two shortest wavelength (highest energy) transitions of the Balmer series. Does that sound right?
So we should maybe think about priorities. I would be interested to hear your opinions. 16 and 17 and 18 are somewhat related to each other and to the new 2D H.O. problem (with the xy perturbation), as well as the H-atom in an E-field problem. (17 includes the infinite sq-well problem from today (and more). 14 and 15 are different.
I would like to hear your opinions regarding priority and order.
To contextualize this a bit, after we do more regarding both kinds of perturbation theory and its effect on the electron states of hydrogen, etc., we will do some variational calculations. These are very important and we will spend some time on that.
At that point we will probably have some time to use these techniques and our related insights to explore a range of physics phenomenology, such as perhaps the orign of the periodic table and the nature of chemical bonding (maybe some other stuff? bonding in metals or semiconductors or something?)
Then toward the latter part of the class, i think we should try to cover understanding how E&M radiation (photons) can induce transitions from one state to another in atoms, and perhaps some scattering theory. WKB theory is not really so interesting or important, so i wasn't thinking of spending much time on that.
I think that maybe we should have 4 problems due at a later date and the rest due Thursday. So maybe 14-17 due Tuesday, and then 18 and the two additional probs due soon after. i know that 14 and 15 aren't so much related to the rest, but maybe it would be good to have a little of both due next class so that we can be more caught up with lecture in a sense. Those are my thought.. anyone else??
sorry, I meant we should have 4 probs due Tuesday and the rest due Thursday.
nina, i always have theta from 0=2pi, and phi from o-pi, like you said, so i agree.
well, it depends on your definitions, but you might approach this by drawing a picture, like with an arm at some angle to the z axis which can then go around in a circle around the z axis (projecting a circle into the x-y plane). Is that the sort of picture you have in mind? It takes two angles to specify the orientation of the arm, i think. Which one are you calling theta and which is phi?
HW problem summary and new problem. Here is my summary of the outstanding HW so far. Comments, corrections, etc, are most welcome and encouraged:
14) 1-D infinite sq. well with delta-funtion: exact solution.
15) 1-D infinite sq. well with delta-function: perturbation calculation.
16) Matrices (2x2), non-degenerate and degenerate (easy).
17) 2D inf. sq. well: degenerate pert. calc.
18) Design your own degenerate pert. problem.
19) 2-D H.O.: degenerate pert. calc. with pert., V'=V0 xy/a^2
20) 2-D H.O.: write simple expressions for the eigenstates of the 2-D H.O.. These can be written simply as products of the 1D HO states, e.g., Psi1(x)*Psi1(y) is the ground state and has an energy hbar*omega. Make sure you are crystal clear on why this is the g.s. and how the excited states can be expressed and organized in this manner. What is the Hamiltonian for the 2D H.O.? What are the excited states and their energies and degeneracies? You can get all this pretty easily from the form of the Hamiltonian and your knowledge of the solutions of the 1D HO. This is relatively easy, but very important as we will use it a lot!
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 x and y raising and lowering operators commute?
In addition to whatever problems end up being due by Tuesday, I would suggest that before next class you look at and think about and maybe at least do a rough draft of the following problems. (These are the problems that are most relevant and important to what we will be doing in class next week, which is all related to degenerate perturbation theory:
16. (foundational, crucial, easy!!!)
20. (foundational, important, not difficult)
21. interesting, useful
19. good problem, there are some insights still to be had here (and the possibility of a gold star!)
17. good deg pert problem
based on the description of each problem, and the context of next week's topics, my opinion for hw due tuesday is 16, 20, 21, 14.
I think that for myself #18 is something I'd really like to think about, and not rush. just my 2 cents.
Zack, would it be feasible for each student to choose the problems to be done by each class, with the understanding that each one has a date that it MUST be turned in by? i.e. the midterm?
Sorry to drag this out so much...but in response to your post, Zack: given an xyz coordinate system you have one angle that varies in the xy plane and one that varies from the positive to the negative z axis. The last variation (r) extends out from the origin.
In terms of which variations are assigned to theta and phi, math convention and physics convention disagree (i.e. the two conventions are the reverse of each other)
Thus in one convention, having a sin(theta)^3 in the integrand would give a factor of 4/3 (integrating from 0 to pi) whereas in the other, it would cause the whole integral to collapse to zero (integrating from 0-2pi).
If you are designing your problem from the beginning, then of course you have the freedom to choose your convention. When given a problem however, it becomes a little more complicated.
So i just looked at hyperphysics.com and it has theta (0-pi) and phi(0-2pi) (physics convention) and wolfram/mathematica (math convetion) says the opposite. This is good for me because it means that my non-zero element for the hydrogen atom perturbation problem will in fact be non-zero.
Regarding the 8 hw problems, do all of them have to be turned in for next week? can we just have maybe 6 of them due tues. or thurs. and dump the other 2 for the following week? it's truly a lot of problems, and i'd like to spend enough time with each of them without worrying about the time constraint for turning them in. from my point of view i'm especially concerned because i have another exam next thurs., so i'd personally like less problems to be due for next week.
Regarding what isotope said I also agree.
For homework, I say that we just group the problems numerically and have those due in groups. I'd prefer to do 14-17 for Tuesday and 18-21 on the following Tuesday. I know this is less than what Zack originally had planned for this Tuesday but we are now venturing out of review homework and it is going to take us (or at least myself) longer to get each homework problem done. Also, I agree with Eliot's opinion in that I want to think about 18 and not rush it.
This schedule unfortunately flies in the face of Zack's list of problems he wants us to consider before next class. I can't see a way to guarantee that we have all seen these problems without making them due that day. We can ask everybody to promise to read them but I don't really trust the effectiveness of this approach over a weekend. Especially if some people have an exam to study for.
Since our lectures are so far ahead of our homework we could do something where 14-17 are due Tuesday still, then we have 19-21 due as a class group quiz by the end of Thursday. This would allow Zack to personally address problems we are having working problems in the form of short lectures as well as alleviate the pressure of having seven problems due in such a short time with a exam (for some of us). How does that sound to everybody?
Dear class,
I'm not entirely sure when to use the perturbed potential and when to use the perturbed Hamiltonian. Or more simply what is the fundamental difference between them? If any.
Elliot's comment focuses on the content of the problems, which is a good approach i think. Here is a possible variation on Elliot's suggestion: what if we do the problems more or less this order: {16, 20, 19, 17} = group 1 (definitely due Tuesday). Then once you finish all those, I would suggest working on 18, or the problem i forgot to post regarding deg. pert. theory for an electron in a electric field....
The advantage that I imagine this approach might have, from a pedagogic point of view, is that it represents total immersion in deg pert theory which might be helpful cause it is new.? I would be interested to learn how long it takes to do each of those problems. Please post regarding how hard they seem to be as soon as you get far enough to have a sense of that. How difficult (or easy) are they?
Regarding 14 and 15, they will relate to things we will do with variational calculations, so unless you have already done them, i think it is okay to wait a bit for those two.
Zack, for 16 part a, I just wanted to ask you if you wanted us to taylor expand for a square root up to second order or to just leave it in square root form i know its probably the former but i just wanted to check just to make sure.
Yes, yes, do the expansion. I should have mentioned that. Good point. And, rereading that problem, i can't understand why the word " exactly" is in it. I think that should have been crossed out. Doesn't it make more sense that way?
PS. no one has responded to captain's "dear class" question, above. (Please respond.)
Dear thephysicist,
the Hamiltonian consists of the kinetic energy plus the potential energy. If you add a pertubation potential to the original potential, then you also add it to the Hamiltonian. Therefore you always add pertubations into the Hamiltonian. The question is only if the pertubation "comes from" the kinetic energy term (like relativistic corrections) or from a potential like the delta-distribution.
Oh by the way: are problems 16, 20, 19, 17 now due to tuesday? I think this is okay!
theephysicist: I'm not really sure what you're having trouble with. As Tim said, the perturbed Hamiltonian is the original Hamiltonian + the perturbation, such as these delta functions we've been dealing with.
However, I have a feeling that this isn't very helpful for you. Can you come up with a more specific situation in which you're confused?
Zack, for 19 do you want us to deal with the first excited states like Psi12 and Psi21, or do you have something else in mind?
Well, that's what I had in mind. Is this too easy?
Maybe for 18 you can come up with something that involves the 2nd excited states?
Also, if this ends up being not quite hard enough, you can also do the -eEx problem (perturbation proportional to x) for the H-atom.
Please don't change x to z, or there is "loss of generality". (Though you could do that as a slightly easier warm-up problem.)
Hi class,
I have a problem with 17. I found the corrections for the energy. Do we need to calculate the 1st order corrections to the wavefunction? If yes, how can I do that? The non-degenerate formula doesn't work anymore since E_n^0 = E_m^0.
This is an interesting question. I am going to step back and encourage your comments on this question. I hope to see some interesting discussion by tomorrow.
Tim, the issue here is that we haven't yet learned a method for calculating the 1st order corrections to the wavefunctions. (is this even something we'll get to?)
In Zack's question, he asks for the 0th order correction to the states. This is because a priori we don't know which basis to use for the degenerate states.
(i.e., for prob.17, the normal basis is Psi(12) & Psi(21). However, any two linear combinations of these two states (provided they are still orthogonal) will also function as an equivalent basis.)
Thus, the 0th order correction to the states is just asking for the which equivalent basis to use.
I agree with Eliot. My understanding of the problem is this: The basis for the subspace associated with the unperturbed hamiltonian is just a diagonalized matrix whose elements are the corresponding energies of the states we are considering.
As Eliot mentioned, any two orthogonal linear combinations of these two states can function as an alternative basis for this system.
One way of finding a suitable set of linear combinations of the states is by considering the subspace formed by the perturbation to the hamiltonian (i.e. the matrix whose elements are the first order energy corrections Psi(nm)|H'|Psi(nm)). If this resulting matrix is not already diagonalized, we can use it to find our new set of linear combinations by solving for the eigenvectors and dotting them with our original basis (Psi(12),Psi(21)).
This new basis is considered a o'th order correction to the wavefunction (something that i still need to think on for a bit).
I'm not sure if this is all valid, but it's what i have been able to absorb over the past week.
word on what Nina said. We are in essence "guessing" that our combination of wavefunctions is an acceptable linear combination, spanning the correct basis, and satisfying the given Hamiltonian. It just so happens to work out that we made an awesome guess in most(some) cases. In some we need to (as Nina said) transform to a more suitable basis.
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