Diffussion of a localized crystal state: electron dynamics in a lattice

This post is a continuation of the "thread" that started with the posts IP#3 and, more recently, "Spec.Prob.#1: subverting the dominant paradigm ".

Based on what we did in class last Thursday, we now have a sense that we can calculate time evolution of a state function without going through the tedious an unintuitive process of doing separation of variables, finding energy eigenstates and then writing the general state function as a superposition. We can instead use an alternate approach based on the formal solution of the time-dependent Schrodinger equation, H Psi(t) = i hbar d (Psi(t)/dt, which is:
Psi(t) = exp[i H t/hbar] Psi(o)
= [1 + i H t/hbar] Psi(o),
where the last line is approximate, but becomes arbitrarily accurate as t gets small.

A key question then, is when might this be useful?

An example of where it might be useful, as we discussed in class, is in the 1D crystal problem, especially the very interesting case in which an electron is initially localized at one site (at t=0) and we would like to understand the time-dependent behavior of the electron, that is, we would like to calculate how the state function of the electron changes as a function of time.

Calculating this time evolution, except for numerical errors and errors associated with the approximations one inevitably makes in dealing with infinite sums, should produce a result mathematically equivalent to a representation of the localized state as a superposition of energy eigenstates (that being the standard approach, the "dominant paradigm").

Looking at the time evolution in this new way, we can answer questions like:
What is the probability that the electron will "hop" from site i to the adjacent site i+1? or a time t?, or, to put it another way:
what is the characteristic time for an electron to hop from one sit to another?

For the two square-well problem, I think that one could get a good estimate of the time for an electron to hop (or "tunnel", as we have said in the past) from one well to another using essentially this method. (This is a problem that we discussed earlier in the quarter, but never actually solved quantitatively.) Things one would like to calculate include the characteristic tunneling time and its' dependence on the distance separating the wells. I think you might be able to do a reasonable "back of the envelope" estimate of that, i think, with the two key things being the wave-function exponential length scale outside the well, the distance between the wells, and perhaps the depth of the well (all of which will go into the overlap integral, right?).
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In addition to being able to do most things one can do with the original methodology based on superposition of eigenstates, this formalism makes it relatively easy (possible) to introduce disorder.
For example, here is a problem involving disorder that might interesting (though, i have never done this before or seen a solution or discussion of this problem. It may be related to "localization".):
Suppose the site at which the electron starts has a different energy than all the other sites. Let say it has a larger negative alpha, and thus a lower energy (a larger, more negative e_0). Then will the electron still diffuse away or will it remain localized?
This is part of the more general question of what effect will disorder have on the propagation or diffusion of an electron in a lattice?

Homework #9

1. For an electron in the ground state of a hydrogen atom potential, calculate the expectation value of:
a) 1/r
b) r
c) x^2
d) the vector r = (x,y,z)
e) p^2
[what is the relationship between the expectation value of r and of x^2 ?

2. a) (Using your results from problem 1) what are the expectation values of the K.E. and potential energy, respectively. Calculate your answer in e.V. and make sure the signs are correct.
How does the K.E. compare with the P.E.? (xc) How does this relate to the virial theorem?
b) Show that the P.E. can be expressed very simply using e^2 and the characteristic length scale, a_0 = hbar^2/m e^2.
c) Show that your result for the K.E. can be expressed very simply using hbar, m and the characteristic length scale, a_0 = hbar^2/m e^2.
d) why does this make sense?

3. For many potentials, there is no intrinsic length scale associated with V(r) and yet there can be quantum length scale which emerges when a particle of mass m is localized by the potential. What are the characteristic length scales for each of the following?:
a) V(x)= -alpha delta(x)
b) V(x) = (1/2) k x^2
c) V(x) = -e^2/r (3D)

4. Using the Psi_n,l,m as a basis, what is the represention of the state Psi = A (x/2a_0) exp[-r/2a_0], where A is a normalization constant independent of r, theta and phi or x, y and z). In other words, write the state Psi = A (x/2a_0) exp[-r/2a_0] as a linear combination of Psi_n,l,m eigenstates. How many states do you need? Is this a mixed state?

5. For the 3D H.O., the 1st excited state manifold contains 3 states: which could be called 100,
010, 001; where 100 includes a term proportional to x, 010 includes a term proportional to y, ...

Show, by construction, that the 1st-excited state manifold of H can be arranged to also include real eigenstate which are proportional to x, y and z (and orthogonal to each other). What else does the 1st excited state manifold of H include? Discuss the differences and similarities of the 1st excited state manifolds of H and a symmetric 3DH.O..

6. a) Use appropriate indentities and relationships to express Psi3,2,2 + Psi3,2,-2 as a relatively simple function of the cartesian variables x, y and z.

b) Show that a 45 degree rotation of this state (around the z axis) will give another state that: is a linear combination of 2 Psi n,l,m states, and is a relatively simple function of x,y and z. What is the linear combination? What would be good names for these states?

7. Use that state from problem 4, let's call it Psi_2x, to make the state (Psi 100 + Psi_2x)/sqrt(2)
a) calculate the expectation value of x for an electron in this state? Is this a mixed state? Is x a function of time?

8. With or without doing any calculations (calculation is extra-credit of you have time), describe the nature of the trajectory of the expectation value of the vector r:
a) in the state: (Psi 100 + Psi 211)/ sqrt(2)
b) in the state: (Psi 100 + Psi 210)/ sqrt(2)

c) Are these qualitatively similar or different? Discuss?

9. a) What is the degeneracy of the 2nd excited state (manifold) of a symmetric 2-D H.O. ?
(extra credit) b) What angular momentum eigenstates are contained within this N-dimensional subspace?

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(extra credit) a) Figure out how to make the state, B (x^2 - y^2) exp[-r/3a_0] from a combination of Psi_n,l,m and/or how to make C x y exp[-r/3a_0].
b) What is the difference (relationship) between these two states? Are they eigenstates or not? Are they degenerate?

Learning and Teaching styles: letter from a former student

I realize that this class can seem frustrating at times. My goal is to teach in a way that is alive and enjoyable, and in a way that can help you learn and remember. The fraction of material that students tend to remember a year or two after a typical physics class has been studied and is surprisingly low. Here is a letter from a former student which I received a few years ago which addresses this point. It is one of a number of communications along these lines I have received from former students.

"Hi, ... I'm at Cornell and I'm finally settling back into school again now that I'm in the second semester. Anyway, I was inspired to send you this greeting right now at this very moment because I'm currently taking Stat Mech and as I was working on my homework for his class, I looked up my old homeworks from Stat Mech back in Santa Cruz. Well, you were my teacher for Stat Mech at UCSC, and of course my teacher for Quantum II, and also my waves teacher for 5B, long, long ago. It just keeps coming up over and over again that those topics are the ones I understand the most! I mean, it just can't be denied that I my understanding is WAY better in those subjects compared to others.

When I was in Quantum last semester and we got into Zeeman effect, changing basis stuff, I had this wonderful, familiar feeling of understanding, and all I had to do was go back and read some of my old homeworks from your class, and then I had it all at my fingertips again. In fact, most of the time at Cornell, I've been feeling pretty far behind all my classmates, but when it comes to the subjects that I studied with you, I actually feel like I am helping them!

When I think of Stat Mech or Quantum II or even 5B, there are only about two or three problems that I really remember from the course, but each of them involved a
serious revelation on my part. I don't actually remember any problems from any other class at UCSC!
... (two paragraphs of examples, etc deleted)...

Wow, you really had a serious impact on my
understanding of physics! Thank you."

Here is one more, that discusses a student's attitude and response, and how that changed over the quarter for this student:

"The first time I took a class from Zack it took me a long time to warm up to his teaching style. He forces kids out of their standard class comfort zone ... of regurgitative lectures and by making them participate and think in class, both of which students really seem to hate. I know i did [hate it] for the first 3/4 of Phys 105. ...
Zack has a very unique teaching style that ... gets to the fundamentals, the very foundations of what he is teaching. ... Student's who take Zack's courses do not just know the material at the end, they have a deep understanding of it...."

Quiz credit...

FAQ:
o So, was the quiz worth anything?
-The point of the quiz was to:
“…help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared”
The real value of the quiz would be if it helped you be ready for and appreciate today’s class, especially the subtle, but important role of the quantized length scale that emerges at the very end of the calculation. It is easy to miss that after the fatigue of a long calculation.

o Can't we leave it at,"a length scale constructed from e^2, hbar, and m is hbar^2/me^2"?
-Yes

o Is there any way we can make up the quiz?
-If you post the answer to the quiz here you will get full credit for the quiz. If you also did it correctly in class you get extra credit.

o "people have been punished directly for showing up late (understandable), and indirectly for finishing homework on time…" -Jon
-Perhaps it might seem that way to you now, but from my point of view no one is being punished. Punishment is not really an issue here at all. This is about trying to create opportunities for you to learn and remember interesting and important physics.

The idea was that if people spent 1/2 hour thinking about length scales and attractive potentials they would get more out of the class.

Homework due dates

How about if we all hand in HW#7 and #8 on Thursday? That way you can all get credit for the work you have done? Would that make people happy? (Or is it too late for that now (: )

Quiz tomorrow (Tuesday)

Instead of actually handing in any homework, how about if we have a quiz tomorrow at the beginning of class. Very short; based on HW8 (which has only one problem).

The quiz will ask for a length scale constructed using only e^2 (in Joules-meters), hbar and m (kg). Any quantity with units of length is correct. Assume e^2 has units of Joule-meters (i.e., so that e^2/r has units of energy). Maybe also a K.E. and P.E.. Those are really easy, right?

Special Problem #1, subverting the dominant paradigm

This is not a problem that you have to do , but, surprisingly, it is not that difficult if you understand everything we have done so far. It takes our study of 1D quantum a step further by approaching the time dependence (of the state function) in a new manner. It also uses matrix formulation. Seeing time dependence in emerge in a different way may provide a new and illuminating perspective and insight into "how QM works".

So far we have also obtained time dependence by: solving the time-independent Schrodinger equation and the writing a given state function, at t=0, as a superposition on energy eigenstates each with their own time dependence. Then the time dependence of the overall state function emerges, somewhat cryptically, from the "de-phasing" of the different exp[-i E_n t/hbar] terms.

But suppose someone came along and said:
Hmm... H Psi = i hbar d Psi/dt ....
what is the difficulty? Isn't the solution just:
Psi(t) = Psi(0) exp[ i H t/hbar]

What would you say to that? Can we apply that in some concrete and useful way???
(Pause here to reflect, argue, consider and react.)

For our crystal problem, consider the state, Psi_j = phi_0 (x-ja), which corresponds to the electron being located (only) at the jth site in the crystal. Consider the set of all such states. In the basis of these states can the Hamiltonian, H, can be written as a matrix? To the same level of approximation we used for IP#3, what is that matrix? (part a))

b) What if you were told (given) that at t=0 an electron is in the ground state of the jth potential. How could you calculate what the electron does as a function of time?

Homework 8 & reading for Tuesday

I know you probably are getting to hate me for posting too much HW, not giving you enough warning, etc, etc., but here is another HW that will help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared: [Comments and questions below would be much appreciated (and will get you credit for this assignment). How about: if you comment, then you don't have to hand it in. How does that seem? PS. These problems are probably more important, right now, than 7 and 8 from problem set #7. ]

** The quarter will be over soon. I think we have only 2 more weeks --4 more chances to meet and discuss quantum mechanics. I hope you don't miss this rare and fleeting opportunity to learn about quantum physics.

1. For a particle of mass m in the potential, V(r) = -e^2/r (in units, as Liboff uses, in which e^2/r has units of energy):
a) What is a characteristic length scale that you can construct using e^2, hbar and m?
b) How do the units of e^2 compare with the units of alpha, where alpha is the strength of a 1D delta function? (comments welcome)
[notice the similarity between the form of the characteristic lengths for this potential, -e^2/r, and the 1D delta function, -alpha delta(x).] (Please comment below)

c) What is a characteristic kinetic energy scale associated with that length scale?
d) For this potential function, what is a characteristic potential energy scale that you can make with that length scale?

2. check later please.

reading: On Tuesday we will go at a brisk pace through the process of presenting the Schrodinger eqn in 3D in spherical coordinates (Del^2) (bring your 3D glasses if you have them), separation of variables, solving for L^2 eigenstates, Ylm, solving the radial equation with the substitution u(r)=rR(r). This is pretty standard and you can read about it in any quantum book if you want to be prepared in that way.
[in Liboff, it is a little scattered: pages 367-380, more or less, treat the theta-phi part and the angular momentum eigenstates; then page 413 shows the Laplacian in spherical coordinates and pages 446 to 449, more of less, cover the hydrogen atom energies, eigenstates, degeneracies, wave function shapes and hybridization*,...

The most interested part will be the way in which we discover the quantization of energy when we solve the radial eqn. We will use the characteristic length scale in that and show how both length scale and energy are quantized together...


* hybridization refers to combining degenerate eigenstates to produce new bases of degeneracy manifolds. This provide a choice of ways to look at and use the eigenstates.

Homework #7

Added problems:
[ note: both 4's are optional. Mike or Kelsey is supposed to post something explaining the problem with 4.o]
4.o (original) Calculate the expectation value of r (the vector), in the mixed state that is an equal mix of the state 00 and one of the (correctly normalized) +-2hbar eigenstates. You know, something like: (20-02 + i 11) or its friend.

4.1 (alt.revised version) It was cool how r went around in circles in problem 3, right? Create a problem in which r goes around in circles that involves a 2hbar eigenstate (of the 2D HO).

(Problems 5 to X are probably more important to understanding H.)

5. Write Del^2 in:
a) 2D
b) 3D

(oh, i mean in both cartesian and in cylindrical (a) and spherical (b) coordinates.)

6. Can part of Del^2 in 5 be expressed in terms of L^2, where L is the appropriate angular momentum "operator"? (you can answer this with a yes or no, and by writing down the part. You don't have to prove it or derive it.)

7. What is the commutator of Lx and Ly? [hint: use the definition of L and the commutators for px and x, etc.

8. Show that if phi is an eigenstate of Lz, then (Lx+iLy)phi is also an eigenstate of Lz.
(most of the time.., i.e., unless it is zero)

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old:
This assignment involves polar plots of 2D H.O. eigenstates (so learning about polar plots is a good idea). If someone would outline a description here, that would be much appreciated.

If you find polar plots confusing, please feel free to do "contour plots" (plots of surfaces of constant value) instead. The main thing is to develop tools to enable you to understand and visualize the nature of a wave-function in 2 and 3 dimensions. (In 1D, we just plotted Psi(x) vs x. In 2 and 3 D we need to work a little harder on visualization techniques...) This will be essential to understanding the H-atom wave-functions.

Do those 1D H.O. states look like correct, normalized states? I think they are all right, but it is always nice to get a 2nd opinion. Problem 3 is time dependent, right?

H atom, electron states in crystals-IP#3

There seems to be a lot of interest in both the hydrogen atom and in electron states in crystals. I think we can do both,and it is true that we can cover angular momentum in the context of studying the hydrogen atom.

Hydrogen will be pretty elaborate: separation of spatial variables in spherical coordinates (we use spherical coordinates because of the symmetry of the potential), solving the theta-phi part (thereby getting angular momentum eig.-states), and then solving the radial part, and finally putting it all together and noticing the unusual degeneracies that occur (s and p states with the same energy...) which are related to less obvious symmetries of the 1/r potential.

With regard to crystal states, I think that some of you will appreciate the opportunity to work on them yourselves, in addition to covering that in class, so I am re-posting both the earlier notes on that and IP#3 here, with the idea class can start in on that as an interactive on-line problem right away. If you work on it i think you would get a much deeper and long-laster appreciation of this important topic, and its beautiful mathematical symmetries.

Here is a suggestion for how to obtain a concrete results. On the last page of the 3-page, handwritten notes, above, there is an expression for Eq. To evaluate that: start by assuming that the denominator is 1 (we can correct that a bit later). Eq is then equal to e0 plus the inner product of phi(x), delta-V(x) and Psi_q (x),

where phi(x) is the ground state of a single, isolated delta-function potential (it should have been called phi_0; and e0 is the energy of that isolated state. (Both of these are well-known to you.) delta-V and psi are both infinite series. That might seem scary, however, try keeping only the 3 terms n=0 and n= +-1 for Psi_q,
and only j=+-1 for delta-V (there is no j=0 term). You will see that this is justified when you do it. Combine complementary terms to get something real, and it will simplify greatly. The integrals are all effortless when you use attractive delta-function potentials!

Please post comments and questions here.

What should we do next?

Please comment here on what you think we should do next. (Please see poll on the right.)

Midterm Problem Poll

A poll regarding which midterm problems you may have liked is to the right. Please feel free to also comment here regarding what problems you liked or disliked, learned from or did not learn from, and why. An image of the midterm is below, for your reference.

My thoughts on preparing for this midterm.

I think that there are different test preparation strategies and that this class is probably at the far end of a spectrum.
In this class there is an emphasis on deep understanding of a few things and mental flexibility and insight, rather than on superficial familiarity with a lot of information. I think it is relevant for studying strategy.

To make a long story short, one can prepare for an informationally intensive exam by studying shortly before and cramming information into your short term memory which you then reproduce on the test. That won't work here since: that information won't be of much value, studying and storing information in that way may reduce you mental agility.

What I would recommend is to study, by working problems, and try understand as much as possible a few days in advance (especially today), and then, by Monday evening, begin to rest and reflect in a restful and not too intensive way. (And get a good night's sleep.) That may help you assimilate your understanding and be prepared to think.

PS. Small details you forget, you can ask for during the test. (But please do prepare a good equation card and do not take advantage of that too much. Mostly, it is to hep you relax and focus on the big picture.) You want to come to this test with your "big mind".

Midterm practice problems

[This is a "live post", meaning i will continue to edit both this top part and add or modify problems.]

The other thing I would mention, is that there are a lot of posts on this blog and it might be valuable for you to take the time to review them all and try to determine: which ones you understand, which ones are most important, and which ones are likely to be relevant for the midterm. Feel free to post questions, a summary, or related stuff here.

Also, on your "card" for the midterm, in addition to the Schroedinger wave eq., p as a derivative, a and a+ in terms as x and p, x in terms of a and a+, and p in terms of a and a+ (and as a derivative wi respt to x (ok, i said that twice), eig states: for inf sq well, 4 eig states for 1D HO, and the ground state for delta function, any thing else?, (and the boundary condition for the delta function) etc., please also include hbar in eV-seconds just in case there is a problem with actual numbers to work out.
I probably left some things out. This list is not comprehensive --just what popped into my head when I was wanting to mention that about hbar. Oh yeah. Length scales for each potential. (or inverse length scales).

Make sure you read all the problems and do as many as you can. You can ask questions.

Practice problems:
1. A particle of mass m (pomm) is in the ground state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? As much detail as you can... what depends on what?...
Perhaps start with a grand statement regarding: what is the central issue of this problem, then get into details and how things depend on other things and so on.

2. A particle of mass m (pomm) is in the 1st-excited state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? same comments as 1.

3. Consider a particle of mass m (pomm) in a 1D harmonic oscillator potential. Suppose that the particle is in a superposition state (mixed state) that is an equal combination of the mth and nth (energy) eigenstates. Calculate the expectation values of: x, x^2, p, p^2, V, T... (anything else?).

4. (More specific). For a pomm in a 1D HO, suppose state is an equal mix of |0> and |1> .
(all states are normalized). a) Calculate the expectation values of x and x^2 and thereby obtain delta x. (i hope this isn't too difficult.) b) Graph delta x (as a function of what?). Discuss, if it is interesting.

5. Suppose a pomm is initially (for t less than 0) in the ground state of a harmonic oscillator potential v=(1/2) k x^2 , and that at t=0 the potential suddenly morphs into a 2-delta function potential with two attractive delta functions of identical strength, one at x = +d/2, the other at x = -d/2 . Discuss what happens, and especially describe what you would find regarding the nature of that state function in the general vicinity of the origin (from like -3d to +3d or so) a long time later!
You may include a discussion of dependencies on strength, separation,...

b) DO the same thing for the case where one delta function is at x=0 and one is at x=+d. Does that make a difference. What might you find to be dramatically different, if anything?

Saturday afternoon additions:

6) Suppose that for t less than zero, an electron is in the ground state of an attractive delta function potential (adp) located at x=0, and that at t=0 two more adp's appear in addition to the first one), one at x= 2/k, the other at x=-2/k.
a) (quick calculation of expectation value at t=0+) Sketch the state function and V(x) at t=0+, and calculate the expectation value of the potential energy of the electron (at t=0+). How does it compare with the value of the expectation value of the potential energy of the electron before t=0?
b) (though question and essay) Describe what happens as a function of time after that. (You may focus on describing the time dependence of the "probability density".)

7. Consider a potential V(x) consisting of 2 identical attractive delta functions. When they are far apart there are two bound states, right? Show that the 2nd bound state ceases to exisit at a specific value of the ratio of the length scales k^-1 and d, where d is the separtation between the two delta functions and k^-1 is the length scale associated with the g.s. of a single delta function potential. (Is this true?) What is that ratio??
b) Describe in words several ways of looking at what happens as you approach this point from the part of parameter space in which there is an odd bound state. (to E, to k ...) (what is happening to the state???

8. (is phase important?)
Suppose you had one ensemble of sq well systems with a pomm in the state (psi_0+psi_1 )/sqrt(2), and another in the state (psi_0 + i psi_1 )/sqrt(2) . Could you distinguish these two cases via energy measurement? explain. how could you distinguish?

9, (phase) For an electron in a infinite square well, discuss the similarities and differences between the state: sqrt(2/L) Sin(2 pi x/L), (the 1st excited state), and the state which is the absolute value of that. Could you distinguish these by energy measurement (of separate ensembles of each)? Discuss that? Are the energy expectation values the same or different? Discuss other similarities or differences and end with a cogent paragraph, imagining that you ar the teacher of 139a, that explains the meaning and significance of phase in QM.
--
10. In 1D QM, which is easier: to be given the potential and asked to find the ground state, or to be given the ground state and asked to find the potential?

oh, i just remembered two more problems (closely related to each other)

11. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite harmonic oscillator (1D).

12. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite square well (1D).
(Is it clear what you are expected to do? Can anyone suggest a better way to phrase these questions?)

Do we need questions in other categories for balance? What are we leaving out?

Key dates: Quiz, Midterm and mid-quarter review problems

Midterm is moved to Tuesday, May 12. We will review "everything" on Thursday (the day after tomorrow).
Mid-quarter review problems 1 and 2 are due Friday at 4:30 PM. You can get extra-credit for doing a good job on these. Asking good questions and helping other people on the blog is also very much appreciated. Working together on groups off-line is also great. Maybe even better?

HW5 1-4, 7 solutions

In equation 24, John mentioned that that actually that should be a plus sign. (Is that what you got?)

In 7, one could also express the expectation value of x^2 in the form (1/2)(b^2) [1 + (t/tau)^2] , where tau is "characteristic time scale". What is that time scale? In thinking about that, one can get a deeper insight if you reflect on the fact that hbar/E has units of time, and fool around with trying E equal to the expectation value of the kinetic energy of an electron in this mixed state.

Homework #5, problems 5,6 &9: solutions and discussion

In 5a, the eigenvectors are obtained pretty quickly by guessing. I wouldn't expect you to converge on the eigenvectors this quickly. Probably several pages of multiplications and guessing to build your intuition is more realistic. Questions 5 and 6 are intended to build your familiarity with matrix multiplication, and eigenvector and e.v. identification, since that will be important in the latter part of this course.

In problem 9, some interesting issues arise. There is a crossover from a region of low transmission at low energy, to a region of high transmission for and incident electron of high-energy. The relationship of that to length scale (wavelength) and energy, and the phase-shift behavior are well worth understanding if you plan to take 139b and/or go to physics grad school. They care a lot about phase shifts and understanding crossover scales there. I only calculated C/A, which represents the transmission coefficient. You can do B/A, the reflectivity coefficient, by the same method and it complements C/A in an intuitive way. (What relation links them?)

Interactive mid-quarter problems

This week I would like you to work on these interactive mid-quarter problems. I believe that these are important problems, and I think will help bring together much of what we have learned and what is important in this class, prepare us for the next stage of this class(1), and apply what we have learned to a very important class of problems(3). I hope you will be able to give them a lot of attention. This is not just another HW assignment, but rather an essential part of the learning and assessment in this class.

There are two contextual factors related to this evolution of format. One is that we are about half-way through the quarter, and i believe that your skills and basic knowledge of quantum physics formalism are developed to a point where it is possible for you to work on more more complex, open-ended and difficult problems. The other is harder for me to explain. But let me try.

In this class, and at a university in general, part of what you learn includes facts, information, methods and formalism. Another crucial part of your education involves problem solving skills, including the ability to evaluate your own progress. Problem solving tends to be straightforward when a problem is clearly defined and you are confident of what you need to do to solve it. It is less straightforward when these things are not clear, when you are wondering: what is this problem is really asking?, what basic method or starting point should I use?, or even what the answer would mean or what form the answer should take.

Yet the skill of working with poorly defined problems --resolving, refining and solving them-- is much more relevant to real world situations, including working at a technology company, doing research in graduate school, and pretty much anything else.

The idea of these problems, is for you to work on them both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. Evaluating your progress can be difficult on your own. There is often the fear that you may be going in completely the wrong direction, and that can be paralyzing. Evaluating direction and progress via discussion with colleagues (other students) is much different from conferring with a professor/expert. For example, you have to evaluate the quality of the advice you are getting, and ultimately, you have to decide for yourself what to do. However, it is also much less challenging than working completely alone. This is an opportunity to develop and demonstrate your understanding key concepts of QM, to learn and to help others.

Interactive Problem #3, bound states of spatially periodic potentials

Crystalline solids, like Cu, Al, Ag, Au, C(diamond), C(graphite), Si, Ge, GaAs, CdTe, NaCl, KCl, LiF, RbI are made of atoms arranged in a crystalline structure. In such a circumstance, when the atoms are relatively close together, the electron states usually are not associated with a particular atom, but rather, extend throughout the crystal. It turns out that one can "build" these extended (non-local) crystal states from the states of individual atoms, and that this is conceptually and computationally a very succesful and useful approach.

Suppose we make a 1-dimensional crystal from an array of delta-function atoms, that is, an infinite array of attractive delta-function potentials of identical strength and equally spaced along the x axis. Using the notes from the earlier post on electron eigenstates in spatially periodic systems, see if you can find the approximate energies of all the electron eigenstates that come from the single bound state of a single, isolated attractive delta-function potential. Perhaps you can identify a "small parameter" that arises and obtain the energies to lowest order in which the energies are not all exactly equal to the starting bound state energy. (Boldness and daring may be called for here.)

b) Is there any redundancy in your states? Define a basis that spans this subspace of states. (Thoughtfulness and careful inspection may be of value here.)

c) Find a good way to present your results regarding the energies of these states. How do they compare to zero? What are possible values of the work-function for this system? How do the energies of the lattice states compare to the energy of the atomic state; how do they depend on "atom spacing", a? and so on.

Interactive Problem #1, mixed states

This interactive question will help us develop a deeper understand of mixed states (and eigenstates).

Suppose you are given 1000 quantum systems all prepared in the same exact mixed state. For concreteness, suppose that: it is an electron in a 1-D harmonic oscillator, and that the coefficients c_0 through c_5 may be non-zero while all other c's are zero. Suppose you want to learn about the energy of the electron and the nature of the state it is in.

What people do in practice to study the energy of electrons in bound state systems is to shine monochromatic UV or x-ray frequency light on them, so that an electron absorbs one photon in an energy conserving process; this give the electron enough energy to become free of the bound state system (i.e., if hf exceeds the "work function"). Then one detects the energy of the free electron (absorbing it into a detector) and inferring the energy of the bound electron state via conservation of energy.

Suppose that you do this with, say, 100 of your 900 mixed-state quantum systems. What will your results be? How would you display them? What would they tell you about the nature of the other 900 quantum systems you have?

PS. It might be useful to make up specific numerical values for the non-zero coefficients, and then discuss things for that specific case. This may help clarify what you are saying to others.

Interactive Problem #2, Lz eigenstates

This problem is designed to help with the transition from 1D QM to dealing with issues that commonly arise in 2- and 3-dimensional QM. This will help you to really understand angular momentum and degeneracy issues when they arise in more complex systems.

As with IP1, the concept and format of this problem, is for you to work on it both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. And then to turn in your own written version of it at the end of the week, which should include a pedagogic description and discussion of your results.

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IP 2.a) Calculate the matrix of the operator Lz in the basis of the 1st 6 energy eigenstates of the symmetric 2D harmonic oscillator. Can you use this matrix to find Lz eigenstates? If so, how? What do you learn from this? Explain. Discuss.****
*****see May 7 comment below

(Why the 1st 6 states?; why not 7 or 5?)
b) Are these Lz eigenstates also energy eigenstates? What energy manifold does each one belong to? Describe the nature and "content" of the degeneracy manifolds of this system. (A nuanced disussion of a is more important than b.)
c) Building on your results from this problem, what other problems can you think of? What are some possible things, related to angular momentum, that one might do next?