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.
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