Today's problem: energies of states in a crystal

So for today's problem, try to finish that by calculating E vs k for that 1-D chain of attractive delta functions and assuming phi_0 is just the g.s. for one delta function. Does that make sense?

Don't forget to graph it; that is important! And think about the energy scale. Is it positive or negative? Are the energies above or below those for the isolated atom? Is Ek real?

Also, you might want to think about the range of k. What if k is 2 pi/a ? or 3 pi / a ? Is there some limit on k or something?

Reflections on QM

Perhaps it would be good for people to share their reflections about QM as comments here. These could focus on what seems most compelling and interested to you, what you would like to remember in 10 years or so or ...?

PS. Please look at the previous post (from today) also.

For Tuesday: more about the periodic table, states in solids, U and the Schro eqn.

So far we have been mostly considering an idealized periodic table, but actually, as some people pointed out, the d-levels don't even start in the correct row, etc. We can discuss this a little more on Tuesday. Perhaps the key things that survive from the idealized picture are the s-p degeneracies (which are v. important!) and the fact that there are always 6 p states, 10 d states, etc. in a group. Anybody have an idea as to why the d-states are shifted out of their expected row? (comments are welcome)

Here is a problem that might be good preparation for understanding electron states in crystals:
Consider a potential that consists of 4 equal-strength and equally-spaced attractive delta functions all in a line (in 1 dimension). Suppose they are relatively far apart*. How many different (orthogonal) low-energy (one-electron) states are there and what do they look like?
(You could warm up with the 2 delta function problem, and then maybe try 3 or 6 delta functions. 3 is a little tricky. *can you define "relatively far apart"?

Is there anything else we should be working on or thinking about for this last week? [After this week it will be just you and the Schrodinger equation* traveling through the world together for the rest of your life. I hope you will be ready for that. (*and a normalization condition)]

Periodic Table, chemical bonding, etc

As we discussed today, the periodic table originated from considerations of bonding, with elements arranged under elements with similar bonding behavior. With the development of quantum mechanics, and, specifically, considerations associated with the degeneracy and angular momentum patterns of one-electron states which are solutions to the Schrodinger equation for hydrogenic atoms (-1/r potential), there is a theoretical basis for understanding the periodic table.

In your assignment, you are asked to explore and establish the nature of the periodic table for harmonic oscillator potentials in 2 dimensions (and 3 dimensions as well if you can). Create a periodic table which might be appropriate for a HO type potential (r^2). Compare and contrast that to the periodic table for -1/r.

Aside from the central issue of the difference in the nature of the periodic table (if they are different), one could also address related intriguing issues such as: How do the degeneracy patterns different? Where do these degeneracies come from? (Why are states of different angular momentum sometimes degenerate?)...

Homework problems to work on ASAP: revised Sunday, more comments please

Many, if not all, of these problems are vague and not clearly articulated. I think they need clarification. Go to it. Clarifying comments and questions are needed, e.g., I think what Zack is asking for here is..., or I don't understand. Are we supposed to ...? That doesn't seem possible, doesn't make sense...

Also through discussion perhaps we can discuss HW priorities and relevance?

I am hoping for and expecting comments from each person actively involved in this class.

41. Suppose a hydrogen atom, for which the electron is initially in the ground state, is perturbed by an oscillating electric field polarized in the x direction, at a frequency which is resonant with the energy difference between the gs and 1st excited states. What are the probabilities, as a function of time, of transitions to any of the 1st-excited states (in the n,l.m basis, unless you prefer another one) and what are the branching ratios (relative probabilities)?
How would the branching ratios change for polarization along y or z?
How could you most effectively stimulate a transition to the 2,1,1 state?
(This is a question, i think, related to the electric field (of the "photon").

42. For variational problems involving two electrons, suppose you write the variational state as the product: Psi_a(r1) Psi_b(r2), where Psi_a and Psi_b are two states that differ in some way, and r1 and r2 are the vector positions of the two electrons, respectively. Discuss both the conceptual and practical significance of symmetrizing the wave function, which can be done via a 2x2 "determinant", i.e.,
Psi_sym = Psia*Psib+Psib*Psia .

You can explore this question with specific examples, e.g., wave-functions for H2 or He. Does symmetrization make a difference in calculating energy expectations values?
Please discuss here...

43. Regarding what we worked on in class today, perhaps you can think of the homework associated with that in terms of the question: Starting from the Schrodinger eqn, and the one-electron orbitals that follow as its solutions (for the -1/r potential): how can we account for and explain the patterns of bonding associated with carbon. Please explain with appropriate math and a few paragraphs of cogent discussion.

44. Consider equal strength delta functions symmetrically arranged around a hydrogen atom center (and of course equidistant from "r=0"). Determine their effect on the degeneracy and order of the 2nd excited states.

45. extra credit review problem: For the 2D harmonic oscillator, analyze the accuracy of the perturbation theory for either: 1) the 4 delta function perturbation from the midterm or 2) the problem with the perturbation V proportional xy problem that we did a long time ago. (hint: i only have some idea of how to actually do something for one of these choices. the other is sort of a decoy. Wouldn't it be funny if that one turned out to be more interesting, haha?

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.

Midterm problems

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.

----

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).

Class Outline

Homework for Tuesday, Oct 21

I hope the GRE's went well for those who are taking them. I am not feeling so well today so any help here would be much appreciated. Does anyone have a good idea of what homework to work on for Tuesday? Thoughts and comments on that, or other stuff, are welcome here!

DegPert Probs17 and 19

Here are some solutions for problems 17 and 19 regarding the effect of perturbations on the 2-fold degenerate 1st-excited state manifolds for the 2D infinite square well and the 2D harmonic oscillator, respectively. These problems tend to start with:
1) establishing a basis of (2) degenerate eigenvectors which span the degeneracy "manifold",
which is followed by:
2) a calculation of the matrix of the perturbation in that basis,
3) determining the eigenvalues and eigenvectors of that matrix,
and
4a) using the eigenvalues to establish the perturbed energies.
Then the last part, which is very important, and conceptually and computationally difficult:
4b) using the eigen-vectors of the matrix to establish "new" spatial (x,y) eigenvectors.

Homework due Thursday, Oct 16

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.

WaveFunctions, 1D H.O. and hydrogen

Hmm. Not meaning to distract attention from the HW post preceding this, but here are some wave-functions for the 1D H.O. and H-atom. These are related to problems 6 and 8 (review).
I am not sure why, but I just LOVE the 1/sqrt(2) under the e{i phi}. How do you feel about it?

They may not be quite right, but let's try to establish and repair any errors through comments here. It is very helpful to have a "standard set" for these as part of our "common culture". If we all use the same, hopefully correct ones, in ongoing and upcoming problems, then, uh, well,... that is a good thing... right?

Homework Problems 14-17 (pert. theory...) and comments

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.)

Brief notes on prob's 1-5

Here are some brief notes on problems 1-5. They mention the discontinuities in the first derivative, and consequent delta functions in the 2nd derivative of Psi, that come up in the T calc's for infinite potentials (sq well and delta function).

A technical question: For web page organization (to avoid post sprawl), I would have rather posted ths as a comment (to the HW 1 post), but i didn't see how to upload an image within a comment. Can I do that? Any advice?

Book for this class; brief outline of what we will cover next.

BTW, there is a book for this class which is Griffiths QM (the same one used in 139a). We will be starting with material similar to Chapter 6 and then Ch 7, i.e., perturbation theory, H-atom fine structure, and variational calculations (that is, calculations of ground states and g.s. energies based on the variational principal). [Ground states are quite important. The idea of a finite kinetic energy ground state (as in the H atom, etc.) is basically a quantum concept with no classical analogue that I am aware of. ]

Regarding perturbation theory, note that there are two very different kinds of perturbation theory, one for non-degenerate states and one for when degenerate states are coupled to each other by a weak interaction term (a perturbation). The lifting of degeneracy and the 0-th order effect on the states is very important and somewhat distinct from the relatively continuous changes that occur in non-degenerate perturbation theory.

Make sure you have a clear idea what degeneracy means in this context.

Homework 1 and welcome to 139b

Hi. This is your first Homework assignement. Welcome to Physics 139b.