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).

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?

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?

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.

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

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

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

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

Your thoughts.

Any thoughts you have on what you would like to learn about or learn more about, please post them here.

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.

Homework 8.

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

Homework 7, answers and some solutions.

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