I am interested in your thoughts a feedback on the final. Did it seem fair, difficult, easy, etc.? Did you like some problems more than others? (see poll on the right.)
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!
Friday, March 21, 2014
Monday, March 17, 2014
Notes on the final (note added wednesday).
Note added Weds, 3-19 9 PM.
By the way, there are 6 problems. Sadly there are none on semiconductors or n-p junction physics. As I mentioned before there is an emphasis on understanding kinetic energy. Square wells or double wells will not be ignored. These problems will provide an opportunity to show your strengths with calculations and concepts. This is an excellent class and I am very impressed with your work, feedback and understanding!
Regarding time management for the final: I think everyone will be able to finish the 6 regular credit problems within our 3 hour time constraint. There are some interesting problems in those 6 that will hopefully challenge you and reveal your solid understanding of quantum physics. People work at different paces and just finishing those 6 is enough, however, if you finish those with time to spare I strongly recommend the extra-credit.
Here is a guide to how to prioritize your time with regard to the extra credit which consists of 3 separate problems labeled: 6c, 7a and 7b. I would suggest prioritizing 6c (extra credit), 7a (extra credit) and 7b (extra credit) in that order. 7a is a "swing for the fences home-run" problem. If you can solve 7a I will be very impressed. If you can come up with something insightful for 6c, that could also be very cool. If you have no idea how to do 7a, then you can just go on to 7b after working on 6c. I hope this helps.
----------------
Your final will cover a number of things, with some familiar problems and other problems that you probably cannot anticipate, but that you will be able to do if you have a deep and close relationship with quantum kinetic energy. I would recommend that in your preparation you cover a range of topics, as we have discussed, and you put some particular emphasis on developing an understanding of quantum kinetic energy. There is a particular emphasis on kinetic energy in the more difficult problems. Also, I believe that there may be a problem that enables you to refer to and use the Schrodinger equation.
There will be no ceiling on this exam. There are 3 serious extra-credit problems at the end that will enable you to soar into the stratosphere if you have the time and inclination.
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 to factor everything with units out and leave behind an integral that will produce a unit-free result, i.e., a number. That way you can check the units of your answer, and observe its dependencies on various parameters, without actually doing the integral. Here is an example:
By the way, there are 6 problems. Sadly there are none on semiconductors or n-p junction physics. As I mentioned before there is an emphasis on understanding kinetic energy. Square wells or double wells will not be ignored. These problems will provide an opportunity to show your strengths with calculations and concepts. This is an excellent class and I am very impressed with your work, feedback and understanding!
Regarding time management for the final: I think everyone will be able to finish the 6 regular credit problems within our 3 hour time constraint. There are some interesting problems in those 6 that will hopefully challenge you and reveal your solid understanding of quantum physics. People work at different paces and just finishing those 6 is enough, however, if you finish those with time to spare I strongly recommend the extra-credit.
Here is a guide to how to prioritize your time with regard to the extra credit which consists of 3 separate problems labeled: 6c, 7a and 7b. I would suggest prioritizing 6c (extra credit), 7a (extra credit) and 7b (extra credit) in that order. 7a is a "swing for the fences home-run" problem. If you can solve 7a I will be very impressed. If you can come up with something insightful for 6c, that could also be very cool. If you have no idea how to do 7a, then you can just go on to 7b after working on 6c. I hope this helps.
----------------
Your final will cover a number of things, with some familiar problems and other problems that you probably cannot anticipate, but that you will be able to do if you have a deep and close relationship with quantum kinetic energy. I would recommend that in your preparation you cover a range of topics, as we have discussed, and you put some particular emphasis on developing an understanding of quantum kinetic energy. There is a particular emphasis on kinetic energy in the more difficult problems. Also, I believe that there may be a problem that enables you to refer to and use the Schrodinger equation.
There will be no ceiling on this exam. There are 3 serious extra-credit problems at the end that will enable you to soar into the stratosphere if you have the time and inclination.
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 to factor everything with units out and leave behind an integral that will produce a unit-free result, i.e., a number. That way you can check the units of your answer, and observe its dependencies on various parameters, without actually doing the integral. Here is an example:
Tia's office hours will be on Wednesday, 3:30-5PM.
Please note, this will be in Tia's new office: NS2 324.
Saturday, March 15, 2014
Final crib sheet.
Reading the comments below, you will see that we decided to not make any change. You will each bring your own crib sheet. Please put your name on that and hand it in with your final. Your crib sheet is expected to contain basic things like wave-functions, integrals and co-ordinate system relationships. You are expected to solve problems and draw graphs live at the final. Your crib sheet should not contain any solved problems, partial solutions or graphs (only basic stuff).
Friday, March 14, 2014
Final prep problems.
1. Using a variational wave-function (like the one we used in class),
determine the size of an electron in a hydrogen atom, and in an He+
potential (2 protons). How are they different.? Why are they different? Graph U and T in each case and discuss their respective roles in influencing size. [Understanding U and T is the important part of this problem.]
2. What are the 4 states of 2 spins? On what premise (based on symmetry) could you divide them into a group of three states and another of just one state?
3. For 2 electrons in a double well potential,
a) write a spatial state (using on the states A and B that we discussed in class Tuesday) that goes with the spin state \(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\) .
[Hint: try starting with:
\(\psi_{A'} = \frac{1}{\sqrt{1+\delta^2}}\psi_A + \frac{\delta}{\sqrt{1+\delta^2}}\psi_B \)
and making a 2 electron state that "respects the symmetry associated with the indistinguishablility of electrons.]
b) write another spatial state that goes with the spin state \(\frac{1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)\).
c) Why do these spatial states turn out to be different?
d) In what way are they different? That is, how does that difference manifest itself?
e) Discuss the consequences of that difference?
4. a) Show that the kinetic energy for an electron in an infinite-square-well energy-eigenstate has zero uncertainty.
b) Calculate the kinetic energy for an electron in an infinite square energy eigenstate.
5. Calculate the expectation value of the kinetic energy of a (Gaussian) free electron wave-packet. (Do that at t=0 to make it easier.)
6. Calculate the kinetic energy expectation value for an electron in the ground state of:
a) an infinite square well
b) a harmonic oscillator
7. Calculate the potential energy expectation value for an electron in the ground state of:
a) an infinite square well
b) a harmonic oscillator
8. Do not ignore HW problems related to a finite square well. What is the energy of an electron in a ground state of an infinite well that is 3 nm wide. Approximately what is the energy of an electron in a ground state of an finite well that is 3 nm wide (and has several other bound states).
9. Review the 1st excited states of hydrogen. Do a calculation that shows where the maximum of (\Psi_{21x}\) is located. Review hybridization possibilities for the 1st excited states of H. What are the essential things that make hybridization interesting; how do they work?
2. What are the 4 states of 2 spins? On what premise (based on symmetry) could you divide them into a group of three states and another of just one state?
3. For 2 electrons in a double well potential,
a) write a spatial state (using on the states A and B that we discussed in class Tuesday) that goes with the spin state \(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\) .
[Hint: try starting with:
\(\psi_{A'} = \frac{1}{\sqrt{1+\delta^2}}\psi_A + \frac{\delta}{\sqrt{1+\delta^2}}\psi_B \)
and making a 2 electron state that "respects the symmetry associated with the indistinguishablility of electrons.]
b) write another spatial state that goes with the spin state \(\frac{1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)\).
c) Why do these spatial states turn out to be different?
d) In what way are they different? That is, how does that difference manifest itself?
e) Discuss the consequences of that difference?
4. a) Show that the kinetic energy for an electron in an infinite-square-well energy-eigenstate has zero uncertainty.
b) Calculate the kinetic energy for an electron in an infinite square energy eigenstate.
5. Calculate the expectation value of the kinetic energy of a (Gaussian) free electron wave-packet. (Do that at t=0 to make it easier.)
6. Calculate the kinetic energy expectation value for an electron in the ground state of:
a) an infinite square well
b) a harmonic oscillator
7. Calculate the potential energy expectation value for an electron in the ground state of:
a) an infinite square well
b) a harmonic oscillator
8. Do not ignore HW problems related to a finite square well. What is the energy of an electron in a ground state of an infinite well that is 3 nm wide. Approximately what is the energy of an electron in a ground state of an finite well that is 3 nm wide (and has several other bound states).
9. Review the 1st excited states of hydrogen. Do a calculation that shows where the maximum of (\Psi_{21x}\) is located. Review hybridization possibilities for the 1st excited states of H. What are the essential things that make hybridization interesting; how do they work?
Thursday, March 13, 2014
Spin video: the role of electron spin in a two-electron state. (symmetry, fermions and all that)
Please post comments and questions here.
Andrew Hudson's question: "You said in the video that delta is proportional to the inverse of the Coulomb force, and it makes sense to have some sort of correction factor for the wave spilling over into the other well for the Psi plus state, but I'm still kind of unclear as to what delta is representing here."
What delta represents is, as you say, the ability of the electron wave to spill over into the other well. To get a perspective on delta, what it means and represents here, let's go back to the case where there is no coulomb force (between electrons). In that case delta is equal to 1. That will lead to the familiar \(\frac{1}{\sqrt{2}}\) factor and a state that has equal weight in either well. Does that make sense?
In the context of this video, we are in a very different regime, where coulomb repulsion is fairly strong and delta is about .3 to .001, roughly speaking. When delta is .3, then about 10% of the probability density is associated the secondary well, and about 90% with the primary well, so mostly the state has the electron in a particular well, but it allows some freedom to extend into the secondary well (for that state). What delta represents is the freedom for the electron to not be completely constrained to be only in one well. It is a relaxing of constraint toward the more general state, e.g., \(\psi_{A'} = \frac{1}{\sqrt{1+\delta^2}}\psi_A + \frac{\delta}{\sqrt{1+\delta^2}}\psi_B \). Does that make sense?
Andrew Hudson's question: "You said in the video that delta is proportional to the inverse of the Coulomb force, and it makes sense to have some sort of correction factor for the wave spilling over into the other well for the Psi plus state, but I'm still kind of unclear as to what delta is representing here."
What delta represents is, as you say, the ability of the electron wave to spill over into the other well. To get a perspective on delta, what it means and represents here, let's go back to the case where there is no coulomb force (between electrons). In that case delta is equal to 1. That will lead to the familiar \(\frac{1}{\sqrt{2}}\) factor and a state that has equal weight in either well. Does that make sense?
In the context of this video, we are in a very different regime, where coulomb repulsion is fairly strong and delta is about .3 to .001, roughly speaking. When delta is .3, then about 10% of the probability density is associated the secondary well, and about 90% with the primary well, so mostly the state has the electron in a particular well, but it allows some freedom to extend into the secondary well (for that state). What delta represents is the freedom for the electron to not be completely constrained to be only in one well. It is a relaxing of constraint toward the more general state, e.g., \(\psi_{A'} = \frac{1}{\sqrt{1+\delta^2}}\psi_A + \frac{\delta}{\sqrt{1+\delta^2}}\psi_B \). Does that make sense?
Student researcher(s) for Topological Insulator project.
I am looking for a student (or students) capable of serious independent research to join a research project calculating the electron states of topological insulators on a pyrochlore lattice.
Monday, March 10, 2014
Spin states. Notes from 3-11 added.
For the material we cover Tuesday (see the post below), spin will play a critical role. At the enclosed link is a summary of the states of two spins (of two electrons). Familiarity with these states, especially the spin state of two electrons:
\(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\)
will help you follow our lecture/discussion on Tuesday.
https://drive.google.com/file/d/0B_GIlXrjJVn4a18zY3pscGZDRzg/edit?usp=sharing
Spin states play a huge role in quantum physics in general (and in quantum computing in particular). Here is the key thing: because electrons are Fermions, their overall state (spatial & spin) is required to be antisymmetric (with respect to the exchange of two electrons). When the spin state fulfills that requirement, the spatial state will be symmetric. When the spin state does not fulfill the anti-symmetry requirement then the spatial state must be anti-symmetric. This can make a big difference in the spatial state and thereby dramatically effect the nature and energy of the ground state.
With regard to quantum computing, for example, in a double-well-qubit the spins states: \(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\) and \(\frac{1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)\) are regarded as the canonical "0" and "1" states of the qubit.
Please feel free to post questions and comments here.
-------
Added notes from Tuesday class, 3-11-14:
Also,
https://drive.google.com/file/d/0B_GIlXrjJVn4N3VkZ2dhYU00cWs/edit?usp=sharing
\(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\)
will help you follow our lecture/discussion on Tuesday.
https://drive.google.com/file/d/0B_GIlXrjJVn4a18zY3pscGZDRzg/edit?usp=sharing
Spin states play a huge role in quantum physics in general (and in quantum computing in particular). Here is the key thing: because electrons are Fermions, their overall state (spatial & spin) is required to be antisymmetric (with respect to the exchange of two electrons). When the spin state fulfills that requirement, the spatial state will be symmetric. When the spin state does not fulfill the anti-symmetry requirement then the spatial state must be anti-symmetric. This can make a big difference in the spatial state and thereby dramatically effect the nature and energy of the ground state.
With regard to quantum computing, for example, in a double-well-qubit the spins states: \(\frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow)\) and \(\frac{1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)\) are regarded as the canonical "0" and "1" states of the qubit.
Please feel free to post questions and comments here.
-------
Added notes from Tuesday class, 3-11-14:
Also,
https://drive.google.com/file/d/0B_GIlXrjJVn4N3VkZ2dhYU00cWs/edit?usp=sharing
Sunday, March 9, 2014
Tuesday: 2-electron states, entanglement and qubits.
For Tuesday's class, based on your feedback and also consideration of what may be perhaps interesting and relevant in contemporary physics, I am thinking that we could discuss: states of 2 interacting electrons (including spin and e-e interaction), quantum entanglement, and the "double-dot qubit". Along the way we may encounter the origins of anti-ferromagnetism and high temperature superconductivity, as well as Hund's rules (electron-electron interaction and correlation play a role in all of these). We may also touch on the concept of broken symmetry and "More is different" (P. W. Anderson, '72). There is a handout in the top post summarizing what you need to know about spin states and spin state notation before our next class.
Update: Looking over the notes I prepared, I am starting to get cold feet. This material looks challenging; it is at a rather high level for 101B. Maybe we should reconsider and do something more ordinary? Please let me know what you think. I may work on a plan B.
* For entanglement, this site may be ok.
http://physics.stackexchange.com/questions/17628/quantum-computing-qubit-creation-entanglement?rq=1
Update: Looking over the notes I prepared, I am starting to get cold feet. This material looks challenging; it is at a rather high level for 101B. Maybe we should reconsider and do something more ordinary? Please let me know what you think. I may work on a plan B.
* For entanglement, this site may be ok.
http://physics.stackexchange.com/questions/17628/quantum-computing-qubit-creation-entanglement?rq=1
Saturday, March 8, 2014
Homework 8 solutions.
From one of your fellow students: comments, commentary and questions and corrections are welcomed.
https://drive.google.com/file/d/0B_GIlXrjJVn4bm9yQjVVeWRhMEU/edit?usp=sharing
https://drive.google.com/file/d/0B_GIlXrjJVn4bm9yQjVVeWRhMEU/edit?usp=sharing
Thursday, March 6, 2014
Cosmo Club (from Tia Plautz)
On Monday (March 10) at 12:30,
the Cosmo Club talk will be given by my mentor of almost 10 years. The
abstract for the talk is given below. If you are interested in
astrophysics or cosmology, I highly recommend you attend. The talk may
get rather technical, but Adrian is an excellent speaker and a great
teacher!
Tia Plautz
Tia Plautz
Wednesday, March 5, 2014
Your thoughts.
Any thoughts you have on what you would like to learn about or learn more about, please post them here.
Tuesday, March 4, 2014
8:30 PM. Online office hours with blackboard.
Let's meet here at 8:30 PM tonight. You can ask questions here and I will respond in real time on a livestream procastor TV channel at http://www.livestream.com/zacksc
After about 8 PM that channel will show my blackboard. You can ask questions here in the comments. Maybe via the procastor channel as well, I am not sure.
After about 8 PM that channel will show my blackboard. You can ask questions here in the comments. Maybe via the procastor channel as well, I am not sure.
Friday, February 28, 2014
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.)
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.)
Tuesday, February 25, 2014
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
https://drive.google.com/file/d/0B_GIlXrjJVn4bl9qd3UxTV9BZ1U/edit?usp=sharing
Saturday, February 22, 2014
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, \(E_k\)?
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, \(E_k\)?
Wednesday, February 19, 2014
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.
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.
Tuesday, February 18, 2014
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
https://drive.google.com/file/d/0B_GIlXrjJVn4NVdPUVlfRk5vck0/edit?usp=sharing
Sunday, February 16, 2014
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.
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.
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.
Friday, February 14, 2014
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?
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?
Tuesday, February 11, 2014
Midterm message from TA Tia Plautz.
I assume everyone got the message from Tia, our TA. In case not here it is:
Saturday, February 8, 2014
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.
Friday, February 7, 2014
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
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
Monday, February 3, 2014
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
https://drive.google.com/file/d/0B_GIlXrjJVn4R0R2eFBsQ2hVZkk/edit?usp=sharing
Sunday, February 2, 2014
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!
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!
Saturday, February 1, 2014
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.
Tuesday, January 28, 2014
Quiz and quiz solutions.
Here is a link to the quiz and below that are my solutions to problems 1 and 2. (These are intended to be a model for you in terms of how much work they show, presentation... Also, I an interested in your comments, thoughts or questions about the quiz. Please do comment here if you have any thoughts or feelings about the quiz (or the class in general). Everything is just below the break.
Saturday, January 25, 2014
Homework 3 solutions link.
https://drive.google.com/file/d/0B_GIlXrjJVn4aGZWMlppVGdLcmc/edit?usp=sharing
As Eric pointed out, there is a mistake in the solution to problem 7 (and problem 8). For 7, the theta integrand is actually \( sin^3 (\theta) \), which integrates to 4/3. This changes the answer from 1.5a to 3a. It also changes the answer for problem 8 to \( 2 \sqrt2 a \), which is about 2.8a, i.e., a little less than 3a.
As Eric pointed out, there is a mistake in the solution to problem 7 (and problem 8). For 7, the theta integrand is actually \( sin^3 (\theta) \), which integrates to 4/3. This changes the answer from 1.5a to 3a. It also changes the answer for problem 8 to \( 2 \sqrt2 a \), which is about 2.8a, i.e., a little less than 3a.
Quiz on Tuesday.
The best way to prepare is to pick a HW problem and see if you can do it on just a blank piece of paper, not looking at any solution. Reading solutions, or any reading, is not as good, except as a prelude to actually working a problem on a blank piece of paper (with your notes as your only aid).
Homework 4. Due Monday Feb 4 (3 PM).
(energy, oscillators, hybrids and L)
Energy:
1. a) Calculate the expectation value of 1/r for an electron in the state \( \psi_{21z} \).
b) Use that to obtain the expectation value of the potential energy for an electron in this state.
c) evaluate that in eV.
d) using your knowledge of the total energy, infer the kinetic energy for an electron in this state.
2. a) Do the same thing for an electron in the state \( \psi_{200} \).
b) Summarize your results in one or two cogent sentences. Include a comparison to what you already know about the ground state.
Energy:
1. a) Calculate the expectation value of 1/r for an electron in the state \( \psi_{21z} \).
b) Use that to obtain the expectation value of the potential energy for an electron in this state.
c) evaluate that in eV.
d) using your knowledge of the total energy, infer the kinetic energy for an electron in this state.
2. a) Do the same thing for an electron in the state \( \psi_{200} \).
b) Summarize your results in one or two cogent sentences. Include a comparison to what you already know about the ground state.
Midterm is on Thursday, Feb 6.
In response to the first comment below, the goal of this midterm is to test, bring into focus and solidify, and perhaps enhance your understanding of quantum physics in the context of our 1D systems and the ground state and 1st excited states of the H atom electron. (The latter goals are just as important as the first.)
A page with states and integrals, as well as "constants" that will help you calculate things in eV, nm type units will be essential. Just one page. No worked problems.
For the 1D HO and infinite square well, bring 3 states each. For the H atom, the gs and 1st excited states will be needed. There may also be finite square well questions. A good qualitative understanding of the finite square well is of value.
One can expect that there will be calculational problems, e.g., expectation values, numerical evaluation (getting something into eV, for example) and qualitative discussion.
I'll add more here as I think of it. Reworking HW problems is an excellent way to start preparing. This web site contains a lot that will help you. Reading a book is not generally an effective way to prepare.
The best way to prepare is to pick a HW problem, or a problem of your own design and of reasonable difficulty, and see if you can do it on just a blank piece of paper, not looking at any solution. Reading solutions, or any reading, is not as good, except as a prelude to actually working a problem on a blank piece of paper (with your notes as your only aid).
A page with states and integrals, as well as "constants" that will help you calculate things in eV, nm type units will be essential. Just one page. No worked problems.
For the 1D HO and infinite square well, bring 3 states each. For the H atom, the gs and 1st excited states will be needed. There may also be finite square well questions. A good qualitative understanding of the finite square well is of value.
One can expect that there will be calculational problems, e.g., expectation values, numerical evaluation (getting something into eV, for example) and qualitative discussion.
I'll add more here as I think of it. Reworking HW problems is an excellent way to start preparing. This web site contains a lot that will help you. Reading a book is not generally an effective way to prepare.
The best way to prepare is to pick a HW problem, or a problem of your own design and of reasonable difficulty, and see if you can do it on just a blank piece of paper, not looking at any solution. Reading solutions, or any reading, is not as good, except as a prelude to actually working a problem on a blank piece of paper (with your notes as your only aid).
Thursday, January 23, 2014
Problem 9 & discussion of \( 2sp^2 \) states.
Hybridization of 1st excited states can lead to what are called \( sp^2 \) states. \( sp^2 \) states are hybrids of the 2px, 2py and 2s states (as in problem 9). These states are orthogonal and normalized, and they are part of an orthonormal basis of that spans the space of all 1st excited states of this potential.
Wednesday, January 22, 2014
\( sp^2 \) -related structures.
These are some important structures in which \( sp^2 \) bonding plays a significant role:
Also, here is a snippet from an intriguing site about bonding. What we are doing is the "fairly high-powered stuff on the wave nature of electrons".
http://www.chemguide.co.uk/atoms/bonding/doublebonds.html#top
Also, here is a snippet from an intriguing site about bonding. What we are doing is the "fairly high-powered stuff on the wave nature of electrons".
http://www.chemguide.co.uk/atoms/bonding/doublebonds.html#top
Problem 10 note.
For problem 10 (HW3) you will need to use del^2 in spherical coordinates to calculate the K.E.. Since Psi100 does not depend on theta or phi, I think you can use the r part which I believe is:
$$ (1/r^2) \frac {d} {dr} (r^2 \frac {d \psi_{100}} {dr}) $$
$$ (1/r^2) \frac {d} {dr} (r^2 \frac {d \psi_{100}} {dr}) $$
Contour Plots.
Here are two examples of contour plots of 2p states created using Wolfram Alpha.
Several things to notice:
1) for the x-y plane (where z=0) I write r as sqrt(x^2+y^2),
2) I left out the "a"s. That is, I set a=1. (so everything is in units of .053 nm)
3) The range is specified. (That is what the x, -6, 6, y, -6, 6 part does.)
(These are not off-center hybrids. They are pure p states; the second one is rotated 45% via a superposition of x + y.)
Several things to notice:
1) for the x-y plane (where z=0) I write r as sqrt(x^2+y^2),
2) I left out the "a"s. That is, I set a=1. (so everything is in units of .053 nm)
3) The range is specified. (That is what the x, -6, 6, y, -6, 6 part does.)
(These are not off-center hybrids. They are pure p states; the second one is rotated 45% via a superposition of x + y.)
Sunday, January 19, 2014
H-atom states and linear algebra concepts.
This post shows wave-functions of an electron in a hydrogen atom potential. In particular, it shows ground state and 1st excited state wave-functions.
This video discusses linear algebra concepts related to question 9 from HW1 and relevant to understanding electron excited states in a hydrogen atom. These concepts include: orthogonality, linear independence, spanning, basis and orthonormal basis.
This video discusses linear algebra concepts related to question 9 from HW1 and relevant to understanding electron excited states in a hydrogen atom. These concepts include: orthogonality, linear independence, spanning, basis and orthonormal basis.
Saturday, January 18, 2014
In Memoriam: Ivan Mercado
Although I barely know any of you yet, in a sense we are all part of a
community -a UCSC community and the community of people who teach
physics and study physics at UCSC. It is with great sadness that I
learned that Ivan Mercado, a member of our community and a close friend
to many of you, passed away tragically and unexpectedly last weekend.
Please feel free to share comments, thoughts and feelings related to
Ivan here.
Wednesday, January 15, 2014
Homework 3. due Friday, 2 PM.
1. For an electron in the ground state of a hydrogen atom:
a) calculate the expectation value of \(r^2\) . [This requires an integral over all 3D space. Don't forget to include the \( 4 \pi r^2 \) factor and to understand where it comes from.]
b) Calculate the expectation value of the potential energy.
c) Evaluate the expectation value of the potential energy in eV. What is its sign?
d) Think about the relationship between PE, KE and E for this state. Since you probably already know that the energy is -13.6 eV (1 Rydberg), can you guess what the KE must me?
a) calculate the expectation value of \(r^2\) . [This requires an integral over all 3D space. Don't forget to include the \( 4 \pi r^2 \) factor and to understand where it comes from.]
b) Calculate the expectation value of the potential energy.
c) Evaluate the expectation value of the potential energy in eV. What is its sign?
d) Think about the relationship between PE, KE and E for this state. Since you probably already know that the energy is -13.6 eV (1 Rydberg), can you guess what the KE must me?
Monday, January 13, 2014
Homework 2 solutions link
https://drive.google.com/file/d/0B_GIlXrjJVn4X2ZtWV9ScU05TWs/edit?usp=sharing
Please feel free to discuss here and to ask questions about any problem/solution that you are not clear about.
Please feel free to discuss here and to ask questions about any problem/solution that you are not clear about.
Saturday, January 11, 2014
Homework 2, due friday Jan 17, 2 PM.
This is our last assignment on 1D QM and, as such, it is important to providing you with a foundation to understand the 3D quantum that we will do next. I would suggest starting soon and spending as much time as you can on this assignment.
Whenever units are needed, please use eV and nm as your default units. Problem 5 is pretty difficult, but I think that if you can do it and understand it then you will really learn a lot about the nature of potential and kinetic energy in quantum systems. This should be useful later in this class and, I think, in future quantum physics classes you take.
1. For an electron in the ground state of a 1D harmonic oscillator, calculate the potential energy expectation value.
Whenever units are needed, please use eV and nm as your default units. Problem 5 is pretty difficult, but I think that if you can do it and understand it then you will really learn a lot about the nature of potential and kinetic energy in quantum systems. This should be useful later in this class and, I think, in future quantum physics classes you take.
1. For an electron in the ground state of a 1D harmonic oscillator, calculate the potential energy expectation value.
Friday, January 10, 2014
Finite Square Well Videos.
In class we have discussed the nature of the electron bound states of a finite square well potential. Our favorite well is the 0.613 nm wide well with walls 16 eV high. In the first video I talk about the bound states of this well. The second and third videos show how these states are obtained from the Schrodinger (Wave) equation for this system.
We will refer to finite wells quite a bit. Part of their appeal for us is that a finite well has bound state wave-functions that extend beyond the confining potential (with an evanescent form, i.e., exponentially decreasing to zero) into a region of flat potential (often chose to be zero). This is similar to the electron bound states of a hydrogen atom potential. It means that we can put 2 finite square wells close together to model a molecule and and explore the origins of molecular binding, which are deeply quantum. We can also examine the nature of quantum tunneling phenomena (from one well to another).
The first video is mainly phenomenological (a long word which means we don't derive stuff, we just look at how it appears), so it is a bit less difficult than the 2nd and 3rd. The 2nd and 3rd videos examine how the quantum wave-functions and allowed bound-state energies are obtained.
We will refer to finite wells quite a bit. Part of their appeal for us is that a finite well has bound state wave-functions that extend beyond the confining potential (with an evanescent form, i.e., exponentially decreasing to zero) into a region of flat potential (often chose to be zero). This is similar to the electron bound states of a hydrogen atom potential. It means that we can put 2 finite square wells close together to model a molecule and and explore the origins of molecular binding, which are deeply quantum. We can also examine the nature of quantum tunneling phenomena (from one well to another).
The first video is mainly phenomenological (a long word which means we don't derive stuff, we just look at how it appears), so it is a bit less difficult than the 2nd and 3rd. The 2nd and 3rd videos examine how the quantum wave-functions and allowed bound-state energies are obtained.
HW 1 solution notes.
Enclosed are some solutions notes for homework 1. I have made these a little short and compact in order to fit it onto 4 pages, so these are not model solutions. Your solutions might, ideally, be a bit longer and more informative. For problem 9 I am planning to make a short video to discuss those linear algebra related issues.I am thinking about adding a short video here to discuss the Linear Algebra related issues of Problem 9, which are relevant to understanding H atom states.
Thursday, January 9, 2014
Homework 1 video.
This video show me working problems 1 and 8 from homework 1. Also, by inference, I think it shows you how to do problem 5, which was also very popular in the poll.
Friday, January 3, 2014
About this class.
Physics 101B meets on Tuesday and Thursday at 4:00 PM.
Instructor: Zack Schlesinger, ISB 243, zacksc@gmail.com
Office hours: TBA
TA: Tia Plautz, tiaplautz@gmail.com
Website: http://physics101b.blogspot.com/
Section: Our class section will meet on Wednesday from 12:20 to 1:50 in room ISB 165 or Thiman 1. At the section you can get help with HW problems and concepts.
Homework is due every Friday by 2 PM in the physics mailroom (ISB232). There will be a box in the mailroom for you to put it in (or you can put in in my mailbox). I would suggest budgeting about 15 hours per week for homework for this class. Ideally that might be over several days, giving you time to mull over concepts, and to let things percolate and "sink in".
Exams: Our final is on Friday, March 21st from 12-3:00 PM. Our midterm will be in February, probably sometime between the 4th and 13th.
Grading: Grading uses a weighting of: 20% HW, 35% midterm and 45% Final.
Website: Our class website will be used extensively and is a critical part of this class. A HW assignment (due Friday, Jan 10) and some review materials are posted below this post. Your participation via comments and questions on the website will be noted and appreciated!
url: http://physics101b.blogspot.com/
Content:
We will start with bound states of one-dimensional (1D) systems (square well, harmonic oscillator) with emphasis on wave-functions (of energy eigenstates) and expectation value calculations. This provides essential background for our next topic: the hydrogen atom where our emphasis will again be on wave-functions and expectation value calculations. Our 3rd topic will be spin and quantum statistics. These 3 topics provide the foundation for understanding our next topics: quantum origins of the periodic table, molecular binding and electrons in crystals. Following this will be the study of semiconductor physics, lasers, quantum spectra and other topics in modern physics.
Summary: (not set in stone)
1D quantum systems (week 1)
H atom (week 2 and 3)
Spin (week 3)
Periodic Table (week 4)
Molecules (week 5)
Solids (week 6)
Semiconductors
& Lasers (week 7)
Quantum spectra (week 8)
Notes: The topics of this class are deeply interconnected. To understand the hydrogen atom one needs a basic understanding of 1D quantum bound states and expectation values. To understand the periodic table it is essential to understand the hydrogen atom states and degeneracies, as well as electron spin. Quantum kinetic energy is critical to understanding molecules and solids. Etc. These connections make the beginning of this class particularly important, as it provides the foundation for what follows. Working homework problems will be a critical part of this class. I would recommend looking at each HW assignment soon after it is first posted (Sunday/Monday) and allowing yourself a few days and maybe 10-15 hours to complete each assignment. Homework is the backbone of the class and provides the means for you to develop understanding and to prepare for the exam problems, which will be similar to the HW problems.
This class provides a great opportunity for us to begin to examine and explore the world of quantum physics, particularly physics at the nanometer scale. Quantum physics is probably one of the most important and unusual breakthroughs in the history of science. Before the wave nature of the electron was revealed, that is, that electron behavior follows a wave equation known as the Schrodinger equation, there was no real model or understanding of the nature of an atom, the organizing principles of the periodic table, why atoms stick together to form molecules and solids, superconductivity, lasers, how photosynthesis begins, etc. Quantum physics remains an active area of research, particularly in novel materials where competing interactions between electrons confound theoretical efforts and sometimes lead to amazing and unexpected new phenomena. In this class we will begin the journey to understand this interesting world.
Instructor: Zack Schlesinger, ISB 243, zacksc@gmail.com
Office hours: TBA
TA: Tia Plautz, tiaplautz@gmail.com
Website: http://physics101b.blogspot.com/
Section: Our class section will meet on Wednesday from 12:20 to 1:50 in room ISB 165 or Thiman 1. At the section you can get help with HW problems and concepts.
Homework is due every Friday by 2 PM in the physics mailroom (ISB232). There will be a box in the mailroom for you to put it in (or you can put in in my mailbox). I would suggest budgeting about 15 hours per week for homework for this class. Ideally that might be over several days, giving you time to mull over concepts, and to let things percolate and "sink in".
Exams: Our final is on Friday, March 21st from 12-3:00 PM. Our midterm will be in February, probably sometime between the 4th and 13th.
Grading: Grading uses a weighting of: 20% HW, 35% midterm and 45% Final.
Website: Our class website will be used extensively and is a critical part of this class. A HW assignment (due Friday, Jan 10) and some review materials are posted below this post. Your participation via comments and questions on the website will be noted and appreciated!
url: http://physics101b.blogspot.com/
Content:
We will start with bound states of one-dimensional (1D) systems (square well, harmonic oscillator) with emphasis on wave-functions (of energy eigenstates) and expectation value calculations. This provides essential background for our next topic: the hydrogen atom where our emphasis will again be on wave-functions and expectation value calculations. Our 3rd topic will be spin and quantum statistics. These 3 topics provide the foundation for understanding our next topics: quantum origins of the periodic table, molecular binding and electrons in crystals. Following this will be the study of semiconductor physics, lasers, quantum spectra and other topics in modern physics.
Summary: (not set in stone)
1D quantum systems (week 1)
H atom (week 2 and 3)
Spin (week 3)
Periodic Table (week 4)
Molecules (week 5)
Solids (week 6)
Semiconductors
& Lasers (week 7)
Quantum spectra (week 8)
Notes: The topics of this class are deeply interconnected. To understand the hydrogen atom one needs a basic understanding of 1D quantum bound states and expectation values. To understand the periodic table it is essential to understand the hydrogen atom states and degeneracies, as well as electron spin. Quantum kinetic energy is critical to understanding molecules and solids. Etc. These connections make the beginning of this class particularly important, as it provides the foundation for what follows. Working homework problems will be a critical part of this class. I would recommend looking at each HW assignment soon after it is first posted (Sunday/Monday) and allowing yourself a few days and maybe 10-15 hours to complete each assignment. Homework is the backbone of the class and provides the means for you to develop understanding and to prepare for the exam problems, which will be similar to the HW problems.
This class provides a great opportunity for us to begin to examine and explore the world of quantum physics, particularly physics at the nanometer scale. Quantum physics is probably one of the most important and unusual breakthroughs in the history of science. Before the wave nature of the electron was revealed, that is, that electron behavior follows a wave equation known as the Schrodinger equation, there was no real model or understanding of the nature of an atom, the organizing principles of the periodic table, why atoms stick together to form molecules and solids, superconductivity, lasers, how photosynthesis begins, etc. Quantum physics remains an active area of research, particularly in novel materials where competing interactions between electrons confound theoretical efforts and sometimes lead to amazing and unexpected new phenomena. In this class we will begin the journey to understand this interesting world.
Homework 1: due Friday, Jan 10
Homework is due Friday by 2 PM in the Physics Dept mailroom across the hall from room ISB 231. There will be box to put your HW in from about noon to 2 PM. (If you come earlier or don't see the box, then you can put it in my mailbox.)
Please do not hesitate post questions or comments here. I encourage you to question anything that seems incorrect or unclear to you. Your questions and comments are strongly encouraged and appreciated. I am also interested to get feedback on whether you find some of these problems too difficult or too easy. Feel free to respond to other students questions and comments. Peer-to-peer dialogue can be very valuable.
This assignment focuses on math-related things which will be relevant to something we will cover in the near future. Feel free to use Wolfram-alpha for integrals, help with graphing, etc.. Graphs should be hand sketched --not too big (or too small). (Two relationships that did not make it onto this assignment are: exp(i*theta)= cos(theta) + i sin(theta), and (e^a)*(e^b) = e^(a+b).)
Waves:
Imagine a string attached to fixed posts at either end. The string's simplest motions, are standing waves, called normal modes, in which the string moves at one particular frequency. For problem 1 you are asked to sketch the string displacement profile at a time of maximum displacement (e.g., t=0) as a function of x.
1. a) For a wave on a string with fixed ends at x=-L/2 and L/2, sketch the lowest frequency mode.
b) Write a mathematical expression for this lowest frequency mode (at a time of maximum overall displacement).
c) Sketch the next lowest frequency mode of a wave on a string (with fixed ends at x=-L/2 and L/2).
d) Write a mathematical expression for y(x) for this mode (at maximum displacement).
Graphing: (Things to notice: overall shape, node location, asymptotic behavior. Please label and/or put scales on axes.)
2. (These are relevant to square well states.)
a) sketch a graph of cos(pi x/L) from x = -L/2 to L/2.
b) sketch a graph of sin(kx) from x = -pi/k to pi/k.
c) sketch a graph of cos(.7 pi x/L) from x = -L/2 to L/2. What is its value at x=L/2?
d) sketch a graph of B e^{-(x-L/2)/a} from x= L/2 to (L/2 + 4a). what is its value at x=L/2? At x=(L/2)+2a ?
Please do not hesitate post questions or comments here. I encourage you to question anything that seems incorrect or unclear to you. Your questions and comments are strongly encouraged and appreciated. I am also interested to get feedback on whether you find some of these problems too difficult or too easy. Feel free to respond to other students questions and comments. Peer-to-peer dialogue can be very valuable.
This assignment focuses on math-related things which will be relevant to something we will cover in the near future. Feel free to use Wolfram-alpha for integrals, help with graphing, etc.. Graphs should be hand sketched --not too big (or too small). (Two relationships that did not make it onto this assignment are: exp(i*theta)= cos(theta) + i sin(theta), and (e^a)*(e^b) = e^(a+b).)
Waves:
Imagine a string attached to fixed posts at either end. The string's simplest motions, are standing waves, called normal modes, in which the string moves at one particular frequency. For problem 1 you are asked to sketch the string displacement profile at a time of maximum displacement (e.g., t=0) as a function of x.
1. a) For a wave on a string with fixed ends at x=-L/2 and L/2, sketch the lowest frequency mode.
b) Write a mathematical expression for this lowest frequency mode (at a time of maximum overall displacement).
c) Sketch the next lowest frequency mode of a wave on a string (with fixed ends at x=-L/2 and L/2).
d) Write a mathematical expression for y(x) for this mode (at maximum displacement).
Graphing: (Things to notice: overall shape, node location, asymptotic behavior. Please label and/or put scales on axes.)
2. (These are relevant to square well states.)
a) sketch a graph of cos(pi x/L) from x = -L/2 to L/2.
b) sketch a graph of sin(kx) from x = -pi/k to pi/k.
c) sketch a graph of cos(.7 pi x/L) from x = -L/2 to L/2. What is its value at x=L/2?
d) sketch a graph of B e^{-(x-L/2)/a} from x= L/2 to (L/2 + 4a). what is its value at x=L/2? At x=(L/2)+2a ?
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