Some problems you may have expected were not there. Most notably, a time-dependent expectation value calculation, such as x or p, for a mixed state. The p calculation tends to be a bit messy and long for a timed exam -too much chain rule - though it would be fine for a take home and it is an important problem worth remembering. An x expectation value calculation is an iconic classic and a very important problem. There was one on the test originally, but I took it out in the interest of shortening the test and allowing time for the novel extra-credit. Frankly, it is also a bit of a muddle to grade and hard to distinguish between who understands it and who is "regurgitating" it (as they say). (The one that would have been on the test was a 1D harmonic oscillator in the state: \( \psi = \frac {1} {\sqrt {2}} \psi_{1} + \frac {i} {\sqrt {2}} \psi_{2} \). The "i" on the 2nd state changes things in an interesting way.)
Regarding problems that were on the test, were designed to test understanding more than just training in a few well-trodden familiar areas. I hope they met those goals. I am interested in how you experienced them. 2 and 3 related to square well energy issues. Did those make sense?
4 was an H-atom hybrid state, but emphasized general understanding rather than calculation. 5 was a length scale/confinement-related problem. Was that challenging? interesting? frustrating?
Problem 6 I hoped would be particularly interesting because it reveals that there are two kinds of quantum kinetic energy that an electron can manifest. Did anyone discover anything interesting there?
Problem 7a takes us back to very basic fundamentals. How did that go? Did some of you figure out how to approach that?
How about 7b?
Please feel free to comment here. Your comments are appreciated.
My solutions are below. Oh, I just realized I haven't done 6c and 7 yet. I'll add those tomorrow. You can do them yourself, as follows. For 6c the key thing was that there are two kinetic energies terms. One is the quantum confinement K.E. that we have discussed a lot (that has no classical analogue), the other is the a K.E. related to motion (hbar k)^2/2m (which does have a classical analogue, p^2/2m). The cross term is zero because odd in x. (For bound states there is only the quantum confinement KE. That is why we saw only that one for most of the quarter.)
7a can be solved by using the time-independent Schrodinger equation. Take 2 derivatives of psi and basically you can get U(x) from that. The graph of U(x) for this psi looks really cool!
Extra credit solution images below:
I feel better knowing that you emphasized trying to put problems we might not have expected, because the test was pretty challenging for me. Overall, I think the test was fair, there were some simple parts (questions 1,3,4) and some that were really challenging (5,6). In summary, I'm disappointed in my performance, but I feel really accomplished thinking about everything we've learned in ten weeks.
ReplyDeleteI was completely lost on what the wave function we were suppose to use for number 2. I ended up writing it was in a general form. I never did part b on 2 because I have no wave function to plug into.
ReplyDeleteI felt that I was a little lost on the wave function we were using.
Do you have an idea now what wave-function might be appropriate there (and how to construct it with the boundary condition your were given)?
DeleteI made up some wave function that had the front constant equal to square root ( 1/ (3L) ) . Then I tried making rude it was normalized but I could not get it. Then I just wrote some what I had. Once I tried integrating it, it became very ugly so I wrote the letter "I" for the integral. Then ending part "a" of the problem.
DeleteFor part "b" I did not have any function to plug in the values. I just had an "I" times a constant.
The wave function outside is a simple decaying exponential, remember. It should integrate almost effortlessly. Normalization is sort of a random thing to try. You don't need to because you are given the value at one point already, and it is not possible because you don't know the wf in the well.
DeleteI got caught up in calculating the new wave function problem and the wave packet problem, and that ate up a lot of time and I felt I wasn't able to answer those questions as well as I could have if I didn't have to worry about differentiating/integrating by hand. Other than that I liked the questions, they weren't too easy, but there was a lot of stuff I had to do by hand under a time crunch that I wasn't used to doing before.
ReplyDeleteYes, that can be a problem. Did you see this post about that issue?
Delete"On one problem you may encounter an integral that is not in your tables. Don't spend a lot of time trying to do a completely unexpected integral. (Your effort to do a novel integral will be unappreciated and will likely take time away from more fruitful endeavors.) Instead, the best thing is..."
"I'm disappointed in my performance, but I feel really accomplished thinking about everything we've learned in ten weeks." -anonymous 1 (you are not alone!)
ReplyDeletei think there are a lot of topics so its hard to get a perfect test. A few pretty easy and obvious questions should be in there (ie #1)
I think that making the wave function from boundary conditions was kind of lame. I think i did it right, maybe, but we were always given our bound conditions pretty explicitly.
Also, the "new wavefunction form" really scared me at first. It looked like i got a good answer, but it could be totally wrong.
I'd rather there be semiconductor questions, or questions related to the bands in crystals.
Also, there really should be a class integral sheet. That would make any test way better.
I felt OK with the test, but I really just hope i passed, which is questionable, despite feeling really comfortable with the first 8 weeks of material.
i think it should be more like a test to see if people are completely lost, and should be pretty passable with somewhat minor challenges. (class crib sheet). I think the test should be more conceptual rather than calculations.
Also, I feel pretty mislead by half of the review questions. I spent a lot of time on the spin states (but didnt get very far anyway) and was just really stressed.
I really liked the course as a whole, though. Zack, you definitely make that class way more fun than it really is! Thanks for that. I can tell you really want everyone to be successful and put forth some good effort. I learned a lot, but I wish the course was slightly more defined (and im a little put off by the lack of semi-conductor questions, because I felt really comfortable with those and tested concepts rather than math, and we spent a good amount of time on them.
pass me! <3
It was a great quarter and I think our professor did a really good job teaching.
ReplyDeleteAh, – for the love of God – for 5a all I had to do was think about units! 'Tis a shame, 'tis a shame...
ReplyDeleteUnits, or the relationship between confinement and KE. Either one would lead to the same place (hbar^2/mb).
DeleteThat's what I got! Although I did have some random constants in there, like a 3.
DeleteOverall I feel like the test was pretty fair, but I feel like it lacked a lot of the content that I was expect. Ignoring the semi-conductors (since it was posted that it would not be on the final) it was missing things like state functions with time dependence, interactions with electric fields and light, and sp2/sp3 hybridized states.
It was a shame that semiconductors were not included considering we spent a decent amount of time on them. However, I was glad to see that there were no spin problems. I think I can speak for most of the class when I say that we are still pretty confused about spin. Even after watching the video you posted I'm still not quite sure what it means to have "spin".
Anyways, I had a great time in this class and I feel like I've learned a lot. I look forward to seeing you again next quarter in 156.
@Son Any random constant there is fine.
DeleteI could not think if any semiconductor questions that were not just simple repeating (regurgitation) of what we had done. Maybe that would have been fine. Like sketching an LED or something.
Problem 4 was a hybrid state!
Ah, well I'm no fool to confinement – but I put hbar^2/(8mb^2) simply because I'm a fool of units! (or was for the test at least).
DeleteSemi-Conductors were a cool addition to the class, but we must not deny the fact that this test was about understanding the true nature of quantum physics. Kinetic energy, confinement, potential energy, the nature of expectation values of non-hybrids/hybrids: these are the core ideas that led to the build up of multiple wells -> chrystals -> Semi-Conductors. All I'm saying is that Schrödinger would have approved of the test – this is for sure. Heisenberg maybe not so much ^_^
Here are my thoughts:
ReplyDelete1) Easy
2) [a] I don't remember us ever solving a potential energy expectation value like this in the past, though we definitely could have. I would have been much happier if there had been an equation on the H.O. or 3D Hyd. Atom and we were using the P.E. equations instead of plugging in a known value.
[b] Impossible to do if you messed up [a]
[c] Good conceptual question
3) [a] Easy
[b] Mistook s for x, I don't remember us ever graphing anything like that before, though we definitely could have. Though I do remember us discussing it so that was my fault, I should have been able to graph it.
4) [a] I don't remember us ever having to differentiate such complicated wave functions to find a max, normally we did it graphically in the past. This seemed too difficult in my opinion.
[b] I thought would have been time dependent, not really sure why it isn't. I understand why and are 0. The energy eigenstate question really threw me off, I intuitively wanted to say yes, but I thought we were supposed to calculate something. So I assumed you meant K.E. eigenstate and got caught up trying to differentiate, which didn't pan out well for me.
5) [a] This was an interesting question, I got the right answer besides the factor in front. I tried a long time trying to figure out what the factor would be through a mix of comparing it to the K.E. equation we had seen before (h-bar^2/4ma^2) and some integration methods.
[b] Good conceptual question.
6) [a] Relived to see an integral I remembered, easy.
[b] This question was very confusing. I had seen it before on the practice problems for the final and couldn't figure it out before either. When you set up the integral in class, the weighting factor was multiplied by the wave function, so I figured that we only had to double differentiate the wave function and that turned out to be wrong. Seemed too difficult for this class.
Overall, I enjoy challenges and having one question on a final that I haven't seen before, but I felt like there were too many curveballs that I hadn't seen before on this final. Especially when this is worth 45% of our grade. I am not happy with how I did and I am really hoping that there is a huge curve because otherwise I'm pretty sure I failed the class (even though I had a near perfect score going into it). I would have liked to see some questions on time dependence, n-p junctions, spin, semi-conductors, integrals of hybrid states... things I thought we covered more than what was on the final. But I do think you're a very good teacher and I really enjoyed your class. So basically it comes down to, I'll be happy if I pass otherwise I'll be unhappy.
Thanks very much for your great feedback! Some specific responses follow:
Delete2. "I don't remember us ever solving a potential energy expectation value like this in the past,"
There was a question just like this in HW 2 I think (problem 5 maybe?). I think I suggested reviewing those in the final prep post.
"The energy eigenstate question really threw me off, I intuitively wanted to say yes, but I thought we were supposed to calculate something. "
yes was all you needed to say. nothing to calculate.
6. "..I figured that we only had to double differentiate the wave function and that turned out to be wrong. "
That is not wrong. That is exactly right.
Seems to me you had a lot of good instincts and intuitions!
I think you'll pass. You're anonymous, right? :)
DeleteHaha notoriously anonymous, but for the record my name is James Henrikson. I'm the tall guy with the shaved head and tattoos. I'm a math major so I won't be too crushed if I fail. Physics is just fun. :) Thank you again for a great quarter, I feel like I finally have a decent understanding of quantum physics, which has always interested me. Even though I found out I will never be able to communicate faster than the speed of light :'( vis-à-vis EPR paradox.
DeleteI went into this final with a high A and left feeling like I may not have passed the class. This was really upsetting, especially because I spent a good while studying a felt I had a very good understanding of the material (I could complete the practice problems and the midterm without much trouble). What really threw me off was the discrepancy between the types of questions on the midterm as well as the final review problems and the final itself. I felt like all my studying did me very little good, since there was very little of the new material on the test and lots of guessing/estimating, which I didn't feel like I was taught to do. It is true that if you understand the material very very well you might be able to guess well, but I just don't see how I was supposed to prepare for that test; it was quite a curve ball.
ReplyDeleteI am seriously disappointed with my performance on that test and do not feel it reflects my understanding of the material at all. I know you want us to be able to think on our feet and figure things out based on what we already know, but it seems to me that if a student can do the vast majority of the homework, midterm and practice final problems without much trouble, they should be able to at least pass, if not achieve a decent grade on the final. To me, it seems like several more of those problems should have been at least partially extra credit.
I really liked this class; N-P junctions (solar cells in particular) are probably the most interesting thing I have learned in my 2 years at college, but I walked out of that final feeling very discouraged. I really do hope you curve this test/class, because I do not feel like the final was a fair representation of my knowledge and passion for the material, and I fear that I may not pass this class if you do not.
A curve implies that there is limited number of high grades, which is not the case. I recognize that this is a challenging test that will have a wide distribution of scores. I really, really doubt that you did not pass. My goal is to give as many high grades as I can justify to myself. (Send me an email when you get your grade if you like.)
DeleteThat seems fair. Thanks for the reassurance.
DeleteAfter getting past the first problem, my heart sank. I was not at all prepared to do number 2 because I did not remember how to derive the wave function using the boundary conditions. We did this last quarter in 101A but because I was just so used to being given the wave function in our past homeworks and midterm, reviewing how to find the wave equation just did not cross my mind. I pretty much left 2a and 2b blank. For problem 3, I feel like I can do this now since I'm now at home, stress-free without any testing anxiety. For 4a, I thought it was a good idea to find the x expectation value for some reason but I guess this wasn't the right way to do it. 5 was ok, and 6 was doable since there was a problem similar to this on the final prep problems.
ReplyDeleteOverall, I feel very disappointed with myself and I feel l that I might have to retake this class. I think it was the testing anxiety and the fact that I wasn't studying the material that really tripped me over. I kept worrying that there would be expectation value problems with integrals that I didn't write down on my cheat sheet so I devoted a huge large amount of studying time on this which was a HUGE mistake on my part. Maybe then having a class cheat sheet would have been useful here. What made me feel so comfortable going into the final exam was (as someone has already said here) the fact that I thought the problems on the final were going to be like the problems on the midterm and the final prep problems. I guess all of this confidence that I had was just an illusion. I was hoping there would be problems on semiconductors. For me, semiconductors was the most interesting part of the class and was the reason why I signed up for PHYS 156. Also, if the key to understanding quantum physics was kinetic energy and confinement then I believe we should have went over this in class more
This class has been amazing and I have learned so much physics in this class. Thank you for providing us the opportunity to give you our thoughts on this final exam and for all of our previous midterm/homework. I'm really hoping that I pass this class so I can take 156.
Thanks for the feedback, etc. I bet you did okay.
Delete"if the key to understanding quantum physics was kinetic energy and confinement then I believe we should have went over this in class more"
More? Could we have emphasized it any more? Well, maybe yes, I guess, but I felt like I was talking about they almost every day.
Hey Zack,
ReplyDeleteI thought the exam was exactly what you hinted at a few days prior. Some of the questions were very interesting and difficult. A few brain farts here and there didn't help but overall I thought the exam was interesting (it kept me thinking for the three hours) and I'm glad to have ended the quarter on such a note (regardless of my grade).
Thanks for such an awesome class!
Looking back on the test I can see that all of the problems were pretty fair. After finishing the test I felt somewhat discouraged. I expected something on a level of difficulty somewhat similar to the midterm which was not the case, the final was much harder. All of these problems required serious thought and a level of understanding that was very high, which I think helps you to judge how much we actually learned (which is a good thing). It is just unsettling leaving a test feeling unsure about certain aspects the problems.
ReplyDeleteAll in all I'm very happy with how the class went and I feel like I have a good enough understanding of QM to explain it to someone in an understandable way, which shows that I learned something!
Thanks very much for a fun quarter!
Thanks for the thoughtful comment.
DeleteTrue, it was a difficult test. People had a very hard time with problem 3, which surprised me. I think I could have asked about that in a more leading manner. Except for 6, 4 and 2, there was not a lot of difficult computation (and for 4 and 2, the computation was not very difficult once you understood the question).
The challenge, I think, was understanding what to do and how to approach each problem:
For 4, how to write a 3D state as a function of only x, when y and z are zero, was the key. After that finding the maximum is not so difficult. If one really understands how the graph of psi21x was originally created this would not be too difficult. On the other hand if you just "catalogued" the particular graphs and their "triggers", without really understanding how to do it yourself, then you would be unprepared for a question like this.
For 5, no computation was needed or wished for. The key was an understanding of the relationship between confinement and KE (or just units).
For 2, the simple form of the wave-function outside the well, and that that was all you needed.
Although taking the test may have been stressful, in grading it we experienced the sense that it revealed real understanding. I prefer this to a test where familiar questions "trigger" familiar responses reproduced partly from memory. Learning to think and apply things in new "environments" is a powerful kind of learning and a key thing in physics.
Regarding problem 3.
ReplyDeleteWhat is the gs energy for 2 square wells merged into one well of width 2L?
What is the gs energy for 2 square wells so far apart that the presence of the other well does not matter? (In which case the symmetric and the anti-symmetric states are the same energy.)
(Once you have that, interpolate...