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| classes:2009:fall:phys4101.001:q_a_1009 [2009/10/11 10:40] – yk | classes:2009:fall:phys4101.001:q_a_1009 [2009/10/22 12:21] (current) – czhang |
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| **If you want to see lecture notes, click [[lec_notes]]** | **If you want to see lecture notes, click [[lec_notes]]** |
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| **Main class wiki page: ** [[home]] | **Main class wiki page: ** [[home]] |
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| ==== Spherical Chicken 10/7 12.38 ==== | ==== Spherical Chicken 10/7 12:38 ==== |
| I am sure this is just a misunderstanding of parameters, but when one has the second solution to the delta function wells, the one with the positive peak, and the negative peak... but it was my understanding, that for instance, in the SHO, at the top of hte peak was where the highest probability of finding the particle was. | I am sure this is just a misunderstanding of parameters, but when one has the second solution to the delta function wells, the one with the positive peak, and the negative peak... but it was my understanding, that for instance, in the SHO, at the top of hte peak was where the highest probability of finding the particle was. |
| However, when you have a negative value peak and a positive value peak... how is this different. | However, when you have a negative value peak and a positive value peak... how is this different. |
| The graphs we did today are all graphs of <math>\psi</math>. To get the expected value of the position, or any other measurable quantity, you have to look at <math>\psi^2</math> which gives you nonnegative probabilities everywhere. | The graphs we did today are all graphs of <math>\psi</math>. To get the expected value of the position, or any other measurable quantity, you have to look at <math>\psi^2</math> which gives you nonnegative probabilities everywhere. |
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| ===Spherical 10/7 13:31=== | ===Spherical 10/7 13:19=== |
| of course. I knew it'd be straight forward. I think I'm just used to assuming <math>\psi^2</math> for graphs these days... | of course. I knew it'd be straight forward. I think I'm just used to assuming <math>\psi^2</math> for graphs these days... |
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| ====Daniel Faraday 10/7 12:45pm==== | ====Daniel Faraday 10/7 12:45pm==== |
| It sounds like the Yuichi really wants to know what we think about lecture and what could make it better. It’s not very often that professors ask for input on how to make their class better, so let’s take advantage of this opportunity. | This whole discussion has been moved to [[Q_A_1012]] |
| For me, the class is working well, because the book is really good and the lectures help me clear up my conceptual understanding. | |
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| I think it would help me to have see a prototypical worked example for each section. I wouldn’t need to see the math worked out, but just an outline of how to do it. | === Spherical Chicken 10/7 13:31 === |
| It would also help me to hear more people ask questions. I learn a lot from other people’s questions. | |
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| What do other people think? What’s working and what isn’t? What could make things better? | |
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| Also, if the class is going well, what made our test scores lower than expected? | === David Hilbert's Hat - 15:00 10/7 === |
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| PS I hope it's okay I posted this here, I know it's not exactly a physics question. | === Zeno 9/8 11:30am === |
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| == Spherical Chicken == | === Ralph 10/09 11AM === |
| I agree. I do appreciate so much that Yuichi cares so much about how we're progressing and that we are on track with him .... (after many professors seemed to care sooo little...) | |
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| For me, the ideal of this class is great. I think a lot of us are having a hard time staying up with reading ahead on time, and that kind of impedes the ability of hte class. I'm honestly trying to make an even greater effort to read ahead more, but I think that's one of hte biggest problems. For example when we're starting tri-terms/mid-terms, it's hard to do homework, study for tests, AND read... | === Spherical Chicken 10/8 21:18 === |
| as for this problem, I think sometimes Yuichi would probably do better to just work on explaining concepts, because, at least for today's lecture, I hadn't been able to read enough to have an honest question. (questions, but not ones that haven't been covered in reading). | |
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| The other thing I was thinking about, is I agree with Mr. Faraday, that having one proto-typical problem is very helpful. Perhaps a day or two after discussing a topic Yuichi could post a problem that we could review with. That way we could have the benefit of studying and working on our own, and discussing in class, and then afterwards, tie up any loose ends with what we didn't understand or have inadvertently overlooked with one example. | === Dark Helmet 10/08 23:12 === |
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| I do appreciate that Yuichi is trying to graduate us to being responsible for our own learning, the way we will be in grad school or in doing our own research -- where we don't get spoon fed solutions with each problem. But I don't think a prototype problem would be detrimental to us. | === Mercury 10/09 4:52 am === |
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| == David Hilbert's Hat - 15:00 10/7 == | === poit0009 10/9 9:58 am === |
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| It seems like the plan is that you read the material, get a general grasp of what's going on, and then have the conceptually finer points talked about in class - but when you come across a problem that is very calculation heavy you can use the book or any other outside resource (integral tables for example) to help you out. So far it seems to be working as a mixed bag; if I read and think about it, then it's nice to get the more conceptually tricky things (as is often the case in QM it seems) and I can do the calculation heavy things without too much pain. But if I don't read and am rather lost, then it just adds more confusion. I think it would be somewhat helpful to get a little bit more of the broad view of things in class - so often we've been looking at various potentials but I'm not sure what they can model or how they can be compared to each other, I just know how to find the wavefunction. | === joh04684 10/9/09 10:15 am === |
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| == Zeno 9/8 11:30am == | === Captain America 10/9/09 10:32 am=== |
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| I think it's a great forum discussion topic. I'll admit that I screwed up the test--pretty badly. It wasn't that I didn't know what was going on, it's more that I just choked under the stress of different conditions. My homework has always been nearly perfect and I do it all myself, I read and reread the book and mark up the hell out of it, I take notes in class, review homework and discussion problems, try outside problems, and I spent a week before hand making up note cards of important formulas, identities and results so I'd be okay without a note sheet or book on the exam. When exam time came I felt panicked, as usual, thinking of a dozen things I should have done to prepare better--maybe review more problems and not focus so much on memorizing--if I had more time. When we're in discussion groups we actively interact to solve problems, getting help from the TA; for homework, we have books and notes and TAs and calculators and integral tables and other outside material (and partners for some) to help us understand and solve problems; during the test we were all exposed to a closed environment where we didn't have much time to interpret the meaning of each question, draw on what we know without reference material, plan a strategy and execute a solution (usually cumbersome mathematically and time consuming). The general class performance on the exam shouldn't be very surprising considering the differences in homework and discussion conditions compared to that of the exam, especially in terms of time allotted. I don't know about the rest of you, but it generally takes me anywhere from half an hour to two or three hours to solve and write up each homework problem. Up to half the time is spent interpreting the problem and planning out a solution strategy. In the test it took me most of the hour to realize that the last question was asking me to solve one of our homework problems--and by the time I had interpreted the question I didn't have time to do more than the first few steps. As soon as I walked out of the exam I set to work successfully solving each of the problems and confirmed the solutions with the book, homework or calculator. It took me longer than the exam time and I needed a little bit of reference material, but I did it; and I think most people in the class can, too. | === Ekrpat 12:40pm 10/9 === |
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| There are three ways I can see to solve this issue: one, making exams more like homework, which would require take-home tests and isn't feasible at a school this large; two, we all know what to expect from Yuichi's exams now that we've taken one and can more efficiently and effectively study for the next exam, and perform much better under the exact same conditions; three, same exam conditions but allowing us to use a page of notes or preferably our textbook. Everybody has the same textbook, we all should have read it, we should be making notes and highlighting important results and concepts, and it's what we reference to solve homework problems. Even professors use lecture notes for derivations or example solutions in class. There's a difference, in my opinion, between knowing how to solve a problem using reference tools and solving a problem in addition to memorized material without reference tools. We just have to determine to which standard we'll be held for exams. In the professional field we'll be expected to know formulas and results to a certain extent. We'll definitely be required to know how to solve every problem we encounter; but we'll still be able to reference our textbooks to solve problems. I would personally like to have open book exams, or be able to use a page of notes. I think I could do much, much better. But even if the conditions of next exam are exactly like the first, I know I'll do much better because I now know what to expect and how to study and prepare for it. | ===Hydra 10/9 8pm=== |
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| Sorry for the long-winded response, but in summary: I'd prefer open book exams and the first exam was a learning experience so students now know how to study and prepare. The next exam should go much better for all of us regardless of testing conditions. | ===Andromeda 10/10 9:31pm === |
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| == Ralph 10/09 11AM == | ==== Spherical 10/8 21:26==== |
| Personally, I like how conceptual Yuichi's lectures are. I also like the way that he organizes them by topic. The hardest thing for me is always figuring out how concepts are related, and a lot of professors do a bad job of summarizing this kind of information. One thing that I think might be helpful is if an important or useful relation is given, that it be appropriately emphasized over the rest of what is being given. It's easy to get lost when there are a lot of equations going up on the board. I'm not particularly good at math, especially math I've never seen before, so it's hard sometimes to recognize when one relation is mathematically more powerful or more general than another. Being given some hints about that would be nice (how key concepts connect mathematically, and ways to see how certain relations are more generally applicable or useful than others). | |
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| == Spherical Chicken == | |
| I agree with a lot of stuff Zeno said -- The only comment I wanted to add was -- I'm fearful of allowing our books to be used -- because professors usually feel they have to make up for the text book by making problems even more difficult and complex -- so given we'd still have the same amount of time, I'd rather not have that option. But I agree that being able to solve a problem and being able to pull it out on demand in such a limited time is kind of .... dis-similar to real live demands and what we should be trying to do in the objective of learning material... I'm very much for allowing a limited amount of notes or a card or something -- something that allows helps us remain confident (something I have a horrible problem with in exams, like Zeno) yet is limited enough to show we understand concepts. Like Zeno said, in real life, we are usually allowed a much more flexible amount of time to solve problems, the ability to look up our own notes etc... I don't see that allowing students to write a small note card would compromise the integrity of the standard, or keep us from obtaining the 'independent reasoning' we're trying to learn. | |
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| == Dark Helmet 10/08 23:12 == | |
| I'm in for the notecard too. We would still need to have the ability to know how to apply what we have on our card, we just then wouldn't have to spend time memorizing every equation perfectly. It also helps because it allows the student to put the information down in a way that they are used to and comfortable with. Sometimes the given equation sheets have the same basic equation in a less helpful way. I would also like to second the thought that this book is good. I really enjoy Griffiths writing style, much less dry than some. | |
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| == Mercury 10/09 4:52 am == | |
| I agree that at least being allowed a notecard would be very helpful. We would still have to understand the physics, but it would require less time because we wouldn't be worried about having each equation perfect. I feel like without some material to reference, we'll be more worried about memorization than actual comprehension of the concepts. I do agree with Zeno, though, in that we now know what to expect so hopefully the next exam will go smoother even if we aren't allowed to use some sort of reference material. | |
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| == poit0009 10/9 9:58 am == | |
| I really like a standard notesheet for the class. I think it wouldn't need to be very revealing, just a few of the main equations. I would also like to have a table of relevant integrals for the quiz. Although I think I could have done any of those integrals, the time it takes to think about how to do it, then actually carry out the calculations, takes up an exorbitant amount of quiz time. | |
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| == joh04684 10/9/09 10:15 am == | |
| I agree, I've always found it more helpful to be able to prepare for a test by not having to worry about needing to memorize and keep track of equations. Having a notecard or an equation sheet lets me focus on concepts and physics rather than memorization. I also agree on the integral table idea, at least for basic ones that we regularly encounter. Another idea would be some basic trig identities. | |
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| == Captain America 10/9/09 10:32 am== | |
| Lecture Improvements: I think Yuichi is doing a great job at the lectures so far, and would agree that an improvement would be a general "this is what to look for when solving equations, this is a good method to set it up," for each chapter. | |
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| As for the test I would agree that not having //anything// given for us was a bit difficult. I tried to remember the most simple of equations thinking that that would be the most we would need, but then was surprised by the amount of equations we needed but weren't given exactly. What I'm referring to I guess would be the first problem where we were expected to solve the integral of sin^2 and the last problem where we should have remembered the first derivative with respect to time of psi. I know we should know how to do these, and I do remember how to do them, but these take a lot of time and that is not a good thing during a test. I actually did the integral during the test of the first problem, but as a result I wasted a good 20 minutes before he said we should just guess the value. This took away enough time that by the time I got around to the last problem I only had about 5 minutes left, and I had to try to substitute in the derivatives and I ended up not finishing the problem -- even though it was merely an easier version of a problem we did for the homework. | |
| Getting a notecard would be a great improvement at least for me, plus it is very calming for some reason to be able to bring that in there and know that if you completely freak out, because the professor says "I don't expect any of you to finish this, I made it long on purpose," at least when your mind suddenly blanks and you start to panic you have a small base of knowledge written down to hopefully bring you back from that white panic. | |
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| == Ekrpat 12:40pm 10/9 == | |
| Making points about how the concepts connect to experiments is very helpful. As is pointing out general things we can apply to more than, for example, the few problems we can simulate with a certain potential. I think that a note sheet would be helpful on the next exam, but I am more in favor of a standard note sheet for the entire class. It really doesn't help to have equations crammed on to a note card when you won't be using them, anyways. If the back of the test was printed with equations Yuichi thinks would be useful, that would probably be enough. | |
| ==Hydra 10/9 8pm== | |
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| I don't think the lack of equations was my problem, it was my (in)ability to actually execute the problems. Like any test, I wrote out the equations pertaining to each problem. I had the very basic foundations (both conceptual & mathematic), but was not able to build off of these. Maybe I should just work through more of the problems at the end of the chapter so I get used to manipulating the equations? So hopefully in class we can do more examples, I'm always game for that. | |
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| As far as open book, I will go with a no. I don't think there was any point during the test that I thought the book would help me. If you have the basic equations & know the concepts, the test should be manageable. | |
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| ==Andromeda 10/10 9:31pm == | |
| I have not all these comments but saw some comments about open book exam and wanted to share my point of view. I am totally against open book exams for two reasons: 1)It is very distracting and hard to focus with people flipping through pages and making noises(sorry but it really is) 2)I tend to not really think about the problem in depth and waste time just looking for that one magical equation in the book. | |
| As far as equation sheet goes i think it will be very helpfull to be allowed to have one especially for integral formulas. it is very time consuming trying to do the integral on the test. | |
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| ==== Spherical ==== | |
| About the Delta Function. I guess I just don't feel comfortable with it because I've used it extremely little... but do we never actually define the delta function, besides just the intuitive case? I went and googled it but do we only use it in terms of limits... solving integrals where the limit is a value vs. 0? Is it more like we're borrowing the //concept// of the delta function? | About the Delta Function. I guess I just don't feel comfortable with it because I've used it extremely little... but do we never actually define the delta function, besides just the intuitive case? I went and googled it but do we only use it in terms of limits... solving integrals where the limit is a value vs. 0? Is it more like we're borrowing the //concept// of the delta function? |
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| == liux0756 3:30pm 10/09 == | == liux0756 3:30pm 10/09 == |
| Yes because V=0 when x is not 0, E>0 is for scattering state and E<0 is for bound state. | Yes because V=0 when x is not 0, E>0 is for scattering state and E<0 is for bound state. |
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| | == Blackbox 8:00pm 10/11 == |
| | If I add a little more on the above two opinions,,, |
| | There are two different states which are bound state and scattering state as you know. |
| | In the case of E>0, Energy is always larger than potential energy in the entire location. |
| | Differently with the bound state, a particle can transmit through the position of x=0 due to the potential has negative infinite value at x=0. I guess that's the reason why the scattering state focuses only on when E>0. |
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| ====Hydra 10/9 21:30==== | ====Hydra 10/9 21:30==== |
| Ok, I just need some verification on the delta-function..... It is not so much a "function" but instead a distribution. And we know that a distribution represents a probability..... But since the delta-function is spiked at one point only....does this mean there is only one probable outcome? I apologize for my fragmented question full of dotted pauses..... but I think it effectively reflects my confusion & frustration with this seemingly hand-waving method. It works, but why? | Ok, I just need some verification on the delta-function..... It is not so much a "function" but instead a distribution. And we know that a distribution represents a probability..... But since the delta-function is spiked at one point only....does this mean there is only one probable outcome? I apologize for my fragmented question full of dotted pauses..... but I think it effectively reflects my confusion & frustration with this seemingly hand-waving method. It works, but why? |
| ===David Hilbert's Hat 10/10 9:20am=== | ===David Hilbert's Hat 10/10 9:20am=== |
| I think Griffiths makes a reference to the delta function when it's used as a distribution of a point particle's mass/charge. Everywhere that's not located at the particle it's zero, but at the particle it's infinity; when you integrate over all space it comes out to be 1, because you have one particle. That seems to be the easiest way to see the delta function - it is very peculiar because it is zero everywhere except at one point it's infinity, but it integrates to 1, like a point particle's distribution (which seems a lot more familiar). | I think Griffiths makes a reference to the delta function when it's used as a distribution of a point particle's mass/charge. Everywhere that's not located at the particle it's zero, but at the particle it's infinity; when you integrate over all space it comes out to be 1, because you have one particle. That seems to be the easiest way to see the delta function - it is very peculiar because it is zero everywhere except at one point it's infinity, but it integrates to 1, like a point particle's distribution (which seems a lot more familiar). |
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| ==== Anaximenes - 22:30 - 10/09/09 ==== | ==== Anaximenes - 22:30 - 10/09/09 ==== |
| This question is about problem 2.34. The problem asks us in part c to show that <math>T=\sqrt{\frac{E-V_0}{E}}\frac{|F|^2}{|A|^2}</math>. However, Eq. 2.139 on page 75 says <math>T\equiv \frac{|F|^2}{|A|^2}</math>. That's with three lines, as in identically equal, as in any statement that they're not equal (such as that in the prompt in 2.34c) is incongruent. Now, I remember the professor said in class that <math>T=\frac{|F|^2}{|A|^2} \frac{k_2}{k_1}</math>, but how do we //show// that? We have no definition of T other than the (patently false) one in 2.139. For shame, Griffiths. For shame. | This question is about problem 2.34. The problem asks us in part c to show that <math>T=\sqrt{\frac{E-V_0}{E}}\frac{|F|^2}{|A|^2}</math>. However, Eq. 2.139 on page 75 says <math>T\equiv \frac{|F|^2}{|A|^2}</math>. That's with three lines, as in identically equal, as in any statement that they're not equal (such as that in the prompt in 2.34c) is incongruent. Now, I remember the professor said in class that <math>T=\frac{|F|^2}{|A|^2} \frac{k_2}{k_1}</math>, but how do we //show// that? We have no definition of T other than the (patently false) one in 2.139. For shame, Griffiths. For shame. |
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| | === Yuichi 11:44 10/11 === |
| | OK. Let's start from a simple case when the momentum before and after the obstacle represented by the potential is the same to show that 2.139 is sensible (if you accept the idea that |A|^2 can be interpreted as the probability for the incoming particle, which should be somehow normalized to 1. Even though you cannot really normalize the plain wave, close your eyes to such a detail.*) If |A|^2 represents that the total probability for the incoming particle (integrated over the entire space!) is one, |B|^2 must be related to the similar probability for the reflected wave, and |C|^2 is for the transmitted wave. Is this acceptable? Or more quantitatively, |B/A|^2 is the probability that the particle is in the "reflected" state, and |C/A|^2 is the probability that the particle would be found in the transmitted state. |
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| | Now in a more general case when the momentum changes after the collision with the potential well (or bump). Paying a bit more attention to what |A|^2, |B|^2 and |C|^2 are related to. They are related to the probability DENSITies that the particle is found __in a unit length in the incident, reflected and transmitted states.__ But if you think about how the reflection and transmission is measured, it's not the spacial densities of the particle in the three waves (incident, transmitted and reflected), but temporal densities which is more directly relevant. //i.e.// how many particle (fractional!) is arriving as an incident wave, going out as reflected and transmitted waves __per second.__ When you do this conversion of probability density per length to per second, you will need to consider |A|^2*k_1, etc. Hence the equation described in problem 2.34. |
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| | If you doing like to ignore the "detail" at *, we have to consider wave packets of incident, reflected and transmitted waves. Following the lecture note for 10/9, |
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| | <math>\psi(x) = e^{-ikx}</math> alone for x<0 region and <math>\Psi(x) = 0</math> will not satisfy the Schrodinger equation and boundary conditions. However, |
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| | <math>\psi_I(x) = e^{ikx}+\frac{B}{A}e^{-ikx}</math> for x<0 |
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| | <math>\psi_{II}(x) = \frac{C}{A}e^{ikx}</math> for x>0 |
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| | will for any value of //k//'s. By taking a linear combination of these for different values of //k//'s, the result should still satisfy the Schrodinger Equation. Actually, at this point, we are really thinking about time dependent Schrodinger equation so the time dependence should also be included in the wave function. So we are thinking about |
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| | <math>\Psi_I(x,t) = e^{i(kx-\omega t)}+\frac{B}{A}e^{-i(kx-\omega t)}</math> for x<0 |
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| | <math>\Psi_{II}(x,t) = \frac{C}{A}e^{i(kx-\omega t)}</math> for x>0 |
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| | before taking a linear combination (<math>\omega=\omega(k)</math> is a function of //k// and <math>\omega(k)\sim k^2</math>), and afterward |
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| | <math>\Psi_I(x) = \int \phi(k) [e^{i(kx-\omega t)}+\frac{B}{A}e^{-i(kx+\omega t)}] dk</math> for x<0 |
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| | <math>\Psi_{II}(x) = \int \phi(k) [\frac{C}{A}e^{i(kx-\omega t)}] dk</math> for x>0. |
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| | <math>\phi(k)</math> is such that <math>\Psi_I(x) = \int \phi(k) e^{i(kx-\omega t)} dk</math> describes the wave packet for the incident particle at t<<0 (well before the collision) situated in where? (x << 0). It will pass through the potential regions of x ~ 0 at t ~ 0, and moves on the x >> 0 at t >> 0 like as if this is a free particle. |
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| | You can show (for you to do if you are looking for a challenge) that given this situation described in the above paragraph, that the terms containing "B" is a wave packet located at x >> 0 at t << 0, and x << 0 at t >> 0, while the "C" term corresponds to a wave packet located at x << 0 for t << 0 and x >> 0 for t >> 0 (not dis-similar to the incident wave packet). Note that for t << 0, only the incident wave packet is in the x region which is appropriate for it (//i.e.// x < 0) and the other two wave packets are ghosts existing only in equations, but not in real life (not dis-similar to image charges in E&M, or if you remember something from waves on a string where waves are reflected, where you were probably told that reflected wave come from the point beyond the fix point of the string where waves cannot physically exist). For t >> 0, the wave packet for the incident particle becomes a ghost sitting in the x >> 0 region, while the other two wave packets are real. |
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| | Here are some more challenges for you, but one can show that |
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| | - the total probability for the incident particle <math>\int_{-\infty}^\infty |\Psi_{\rm inc}(x,t)|^2 dx</math> where <math>\Psi_{\rm inc}(x) = \int \phi(k) e^{i(kx-\omega t)} dk</math> remains unity for any time, //t//, once <math>\psi(k)</math> is determined appropriately (so that the total probability is unity at some time t << 0). |
| | - the total probability for the particle associated with the reflected wave <math>\int_{-\infty}^\infty |\Psi_{\rm ref}(x,t)|^2 dx</math> where <math>\Psi_{\rm ref}(x) = \int \phi(k) \frac{B}{A}e^{-i(kx+\omega t)} dk</math> will be |B/A|^2 and remains the same for any time, //t//. For R, the reflection coefficient, one is interested in the total probability in the reflected wave packet for t >> 0 over the incident wave packet for t << 0, which must be 1 if <math>\psi(k)</math> is correctly determined. This is |B/A|^2.\\ \\ To figure out the total probability associated with this wave packet, you need to modify this expression so that you can use the fact that the probability for the incident wave packet integrate to 1 |
| | - Very similar claim can be made for the transmitted part of the waves. Note that in case that the momentum of the transmitted wave is different, "k" in the plane wave expression has to be changed to <math>e^{i(k'x-\omega t)}</math> and the wave packet will look like,\\ \\ <math>\Psi_{\rm trans}(x) = \int \phi(k) [\frac{C}{A}e^{i(k'x-\omega t)}] dk</math>. Note that only "k" in the plane wave is changed to "k'", but not the other. (think about why if you don't want to just accept it from me.)\\ \\ To figure out the total probability for the transmitted wave packet, make a reasonable assumption that <math>\phi(k)</math> has significant non-zero values only for a small //k'// range around it peak at <math>k_0'</math>, which corresponds to <math>k_0</math> for the incident wave. Then one can approximate that \\ \\ <math>k' = k_0'+\frac{k_0}{k_0'}(k-k_0)</math>. I will leave the rest for you to prove. |
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| | The following may be useful for some of the calculations needed to prove some of the things above. When the incident wave packet, <math>\Psi_{\rm inc}(x) = \int \phi(k) e^{i(kx-\omega t)} dk</math>, is properly normalized, <math>\int_{-\infty}^\infty [\Psi_{\rm inc}(x)]^*\Psi_{\rm inc}(x) dx = 1</math>. By substituting the former into the latter, and change one occurrence of //k// to //k_2// to remember that the two integrals involve different variable of integrals, |
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| | <math>\int_{-\infty}^\infty [\int \phi^*(k_2) e^{i(k_2x-\omega_2 t)} dk_2]^*\int \phi(k) e^{i(kx-\omega t)} dk dx = 1</math>, where <math>\omega_2 \equiv \omega(k_2)</math>. |
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| | Since we are sloppy physicists, we freely change the order of the integrals. So rather than doing //k// and //k_2// integrals, we decide to do the //x// integral first. This will result in |
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| | <math>\int \phi^*(k_2)dk_2 \int \phi(k) dk \int_{-\infty}^\infty dx [e^{-i(k_2x-\omega_2 t)} e^{i(kx-\omega t)}] = 1</math>. |
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| | <math>\int \phi^*(k_2)dk_2 \int \phi(k) dk [e^{+i\omega_2 t} e^{-i\omega t}]\int_{-\infty}^\infty dx [e^{-ik_2x} e^{ikx}] = 1</math>. |
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| | Since the most inside integral is one way to express the delta-function (times 2pi) (see Griffiths' hand-wave argument for this), //k//_2 integral becomes easy: just replace //k//_2 with //k//. |
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| | <math> 2\pi \int \phi^*(k)\phi(k) dk [e^{+i\omega t} e^{-i\omega t}] = 2\pi \int |\phi(k)|^2 dk = 1</math> |
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| | This is the normalization equation for <math>\phi(k)</math> similar to what we used to have for "A" in some practice problems, or <math>\sum_n |c_n|^2 = 1 </math>. |
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| | I may have made some mistakes like where 2pi goes, so please carefully check if you are interested in this issue. |
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| | === Can 10/21 10:52am === |
| | For Yuichi, instead of saying A^2 is probability, I think it would make much more sense if we think of A^2 as incident intensity, just my personal opinion not sure if it is rigorous. Recall that the square of the amplitude of for E field is proportional to the intensity , E^2 is proportional to I, same analogy can be made here, then the probability can be interpreted as transmitted intensity versus incident intensity. <math>T=\frac{F^2}{A^2}</math>. At least, I think this might be easier to understand transmission intuitively, maybe not mathematically, one still has to work out all the math. |
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| | ====Liam Devlin 10/12, 10:45am==== |
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| | Why is k a known for a scattering problem? |
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| | ===Can 10/22 12:15pm === |
| | Since for scattering problem <math>E=\frac{h^2k^2}{2m}</math> which means <math>k=\frac{\sqrt{2mE}}{h}</math>, and E is the incident energy of the particle, which can be manually tuned. E is known , so k is known. |
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| | ====Aspirin 10/12, 2:30pm==== |
| | Why does the coefficient, D, of the genernal solution which is <math> \psi_I_I = C e^{ikx} + D e^{-ikx} </math> cancel out? |
| | Yuichi mentioned the reason, but I didn't fully get it. |
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| | ===Blackbox 10/12, 3:20pm=== |
| | I'm not quite sure I understood that correctly, but if I say, when a particle moves from the left to the right direction, it will transmitt through the Delta function well. In other words, there is no physical values from the right to the left direction. If we think of the opposite situation, a particle moves from the right to the left, in the first region, <math> \psi_I = Ae^{ikx} + Be^{-ikx} </math>, A would be removed because nothing can go back to the right direction. |
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| | ====Blackbox 10/12, 6:00pm==== |
| | I might write wrong in my note though. On last Wednesday, Yuichi mentioned the unit of α is Energy/length. Isn't it Energy*length? |
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