Homework 8.

edits: Tuesday 2 PM: Problems 3 and 8.
1. Sketch a picture that shows the conduction and valence bands of an n-p junction as a function of x, the distance from the interface. Show how the bands bend upward in what we call the "junction region", and then level off after that. (FYI, technically, the conduction band in the n region should never be below the valence band in the p region for an ordinary junction. Let that "constrain" your drawing.) 

Homework 7, answers and some solutions.

Here is a link to some answers and partial solutions for HW 7. Please post questions or comments here.
https://drive.google.com/file/d/0B_GIlXrjJVn4bl9qd3UxTV9BZ1U/edit?usp=sharing

Homework 7. extra problem (9) added.

Notes: 7 and 8 were added  Feb 25, 8 PM.  Problem 9 (extra credit) added Feb 27, 9 AM. Also, I made 5 move specific and hopefully more clear.

1.  a) Show that \(\psi_k = e^{ikx}\) is an energy eigenstate of the Schrodinger equation for a free electron:
$$ - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = E \psi(x) $$ What is its energy eigenvalue, \(E_k\)?

Reading: Chapter 10, Modern Physics, Harris.

I would somewhat recommend reading Chapter 10 of Harris, Modern Physics, which cover things like: molecules, crystalline solids, conductors (metals), semiconductors and semiconductor device physics. Skip the section on vibration and rotation. Although we have not been really following the book so far, I think this chapter may be helpful in understanding this material, which encompasses a lot of phenomenology.

Note added Feb. 23. I believe that the book includes 2 approaches to the formation of energy bands in crystals and this can be confusing. One is that each band comes from an atomic energy level and state, the other is the free-electron approach. While both have their uses, the first approach, in which each band arise from a particular atomic state is the better one in my view. I would take anything you read (anywhere) about the free-electron approach to band theory with a grain of salt.
Please post any questions or comments related to the reading here. If something there seems different from what I have said, please bring that up here. If anything seems unclear in general, it is helpful to everyone to bring that up for discussion here.
Over the next week or two we will be covering things like electrons in crystals, semiconductors, semiconductor lasers, leds and solar cells and related topics.

Homeowrk 6 solution notes.

These are solution notes, not model solutions. Please do post any questions here.

https://drive.google.com/file/d/0B_GIlXrjJVn4NVdPUVlfRk5vck0/edit?usp=sharing

HW 5: solution to problems 1-5. And 6.

I added 6 this morning.
Please discuss in the comments here. Is this what you imagined/did for problem 5?
https://drive.google.com/file/d/0B_GIlXrjJVn4aTFSYkhHWUFwdnc/edit?usp=sharing

Problem 5 and 6 are below the break.

Homework 6.

edits:
 5:00 PM, Feb 19: [problem 3 clarified. 7 edited. 7 is interactive.]
11:00 AM, Feb 18: [problem 1 note added for clarity.]
Do all these problems seem worth your while? What is your favorite and least favorite? If you were to eliminate one problem, which one would it be?

[Kinetic energy.] 
1. Consider a 2 square well system: 2 identical wells of width L spaced a distance S apart. If you put the origin of your coordinate system exactly between the wells, then the wave-function in that center region is of the form: \(A_1 cosh(x/a_1)\) for the ground state and \(A_2 sinh(x/a_2)\) for the 1st excited state. You are given that: \(A_1=0.2 (\frac {2}{L})^{1/2}\), \(A_2= (\frac {5}{4})A_1\) and \(a_1= a_2=0.2\) nm, and the region separating the 2 wells is 0.4 nm wide.
a) Sketch the ground state and 1st excited state (over all regions). Pay particular attention to the center region and make your sketch reasonably consistent with the numbers given for the pre-factors and length scales. 
b) For the ground state, calculate the contribution to the kinetic energy integral from just the center region (that is, integrating from x = -0.2 nm to +0.2 nm). (To make your result readable by the graders, and to you, please leave your answer in terms of \(A_1\) and \(a_1\) and a numerical result for the definite integral over \(cosh^2(x/a_1)\), which will have units of nm. (By the way, those nm will cancel the units of \(nm^-1\) from \(A_1^2\) .) There will, of course, be other constants that always are present for a K.E.. )
c) Do the same thing for the 1st-excited state. [Again, please leave your answer in terms of \(A_1\) and \(a_1\) and a numerical result for the definite integral.]
d) Summarize/discuss your results. What does this suggest about the overall K.E. for these states?
e) (extra credit) Plot the integrand in each case; show the area under it and discuss.

Follow-up comments on homework 5.

Homework 5 seemed like quite an endeavor with the a range of questions that may have been resolved or not. I am not sure. I would be very interested in your follow-up comments. That will also help me prepare HW 6.

Did problem 4 work out for you?  Did problem 5 make sense? Were you able to glean anything from the requested graphs (b or c)? Did you resolve what to calculate for 5d? Did that calculation work out okay? Were you able to get a simple result for 6? What was it? Were the issues related to conjugation of \(\psi_{21x}\), \(\psi_{21y}\) and \(\psi_{211}\) worked out to your satisfaction?

Midterm message from TA Tia Plautz.

I assume everyone got the message from Tia, our TA.  In case not here it is:

HW 5, due Friday 3PM. We need more peer-to peer discussion. Please join the discussion (e.g., regarding problem 5).

This assignment looks difficult to me. I would suggest starting very soon. Also, I encourage you to use comments to clarify what is asked in these problems. Especially for something like problem 5, this is a good forum to discuss what one should actually calculate and develop a consensus on that. For problem 5 b) I would recommend factoring out an overall \(e^{-i E_1 t/\hbar}\) factor. That may make it a little easier to see what that graph looks like.

Midterm solutions and commentary.

      A midterm is an opportunity to evaluate, but, more than that, it is also an opportunity to provide emphasis. Realistically, many things are taught in physics classes that are neither learned nor remembered.
      One could ask oneself, what does this midterm communicate regarding what are points of emphasis from the 1st half of this class? What does it communicate about what the teacher may think is particularly important to learn and remember? (more on this below the break)

https://drive.google.com/file/d/0B_GIlXrjJVn4OXgyY2dhOTMzWnM/edit?usp=sharing

Homework 4 solutions link.

Here is a link to HW 4 solutions: In problem 3b, the limits of integration are incorrect. They should be -infinity to infinity.

https://drive.google.com/file/d/0B_GIlXrjJVn4R0R2eFBsQ2hVZkk/edit?usp=sharing

Practice midterm problems & notes added weds 9 pm.

Notes added  (also see below for some added solutions related discussion): In addition to an in-class midterm there will be a short take-home due the night of the midterm.

This is a draft of some practice midterm problems. The level of difficulty here tends to be on the higher side; I would recommend going for mastery of the less difficult problems first. The midterm will not be quite this difficult or this long, but it will be difficult and I expect it to take the full 1hr and 45 min class period.  This is just a bunch of problems without detailed design. The midterm will have some straightforward problems and some more difficult ones. Being comfortable with the straightforward problems is very important!

Homework 4 notes and videos.

Here are some videos that address questions people were having about off-center hybridized states and related matters on HW4. One key thing to realize is that if you do not have states of different symmetry, then you cannot make off-center states (hybrids). Without the radial state, \(\psi_{200} \), one would not be able to make off-center states from the x, y and z states. Regarding symmetry, the radial state is not changed by any rotation. The x, y and z states, on the other hand are invariant only with respect to rotation around one axis (the one in their name), and thus have a different symmetry.