I'll have office hours on Friday 2-3 PM. Hope to see you there! (ISB 243)
With regard to my office hours, I would like to ask a special consideration: because my health and immune system are somewhat compromised, we should avoid close proximity or contact if you might be sick or feel like you may be coming down with something. Thanks very much. Your consideration with this is really appreciated!
Thursday, March 31, 2016
Reorganization of homework 1.
I have been thinking about our first homework assignment. I think I need to be more clear about my expectations and hopes with regard to that. I decided it would not be a good idea to just extend the due date. Instead, I broke it up into parts. I would like you to finish parts 1 and 2 (problems 1-4) by Monday.
Even more importantly, I would like you to start sharing your results and questions on the blog in a timely manner. My hope and expectation is that most of you, unless you are really lost, will begin to comment frequently. Where are you stuck? What are you thinking? Do you have any results to share? Does your result agree with Sophie's? etc., etc. I am hoping to see comments from most of you today.
Even more importantly, I would like you to start sharing your results and questions on the blog in a timely manner. My hope and expectation is that most of you, unless you are really lost, will begin to comment frequently. Where are you stuck? What are you thinking? Do you have any results to share? Does your result agree with Sophie's? etc., etc. I am hoping to see comments from most of you today.
HW1-part 3. states of multiple square wellls.
Please comment here regarding and thoughts, questions, ideas or results you have for these problems.
9. For a system of 4 identical attractive square wells that are equally spaced and pretty close to each other but not touching,
a) sketch the ground state.
b) sketch a few excited states.
c) How did you construct those states? Discuss. What would come next (in terms of states)?
10. a) For a double well (2 identical attractive square wells that are pretty close to each other but not touching), show that for some separations the states come in pairs. What is the nature of those pairs? Discuss qualitative aspects of their construction and energies.
b) What is the analogous thing for a quadruple well (as in problem 8)? Discuss.
9. For a system of 4 identical attractive square wells that are equally spaced and pretty close to each other but not touching,
a) sketch the ground state.
b) sketch a few excited states.
c) How did you construct those states? Discuss. What would come next (in terms of states)?
10. a) For a double well (2 identical attractive square wells that are pretty close to each other but not touching), show that for some separations the states come in pairs. What is the nature of those pairs? Discuss qualitative aspects of their construction and energies.
b) What is the analogous thing for a quadruple well (as in problem 8)? Discuss.
HW1: part2. Bound states of a finite square well.
This is a collaborative problem*. Please share your questions and results here. We want to get to the answers and an understanding of this problem through our collective effort. Posting comments here is my idea about how that collective effort can take place. This is a somewhat difficult problem. Your efforts and comments here will be very much appreciated!
4. Consider a well of the width we found for problem 1, and with a depth of 100 eV:
a) Is the energy of the ground state above or below -90 eV?
b) How many bound states does it have and what are their energies?
c) Sketch the ground state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well? (We can call this the evanescent part of the wf.)
d) Sketch the 1st-excited state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well for this state?
e) Sketch the highest energy bound state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well for this state?
f) extra credit: Create normalized bound state wave-functions for any or all of these bound states. Do a nice graph and maybe compare them to the corresponding bound state from a infinite square well of the same width.)
*The way I envision this is that we work on this problem collaboratively here, freely sharing and comparing our ideas, questions and results. And, in addition, you hand in a nicely organized written solution. Not one or the other, but both. Does that make sense?
4. Consider a well of the width we found for problem 1, and with a depth of 100 eV:
a) Is the energy of the ground state above or below -90 eV?
b) How many bound states does it have and what are their energies?
c) Sketch the ground state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well? (We can call this the evanescent part of the wf.)
d) Sketch the 1st-excited state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well for this state?
e) Sketch the highest energy bound state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well for this state?
f) extra credit: Create normalized bound state wave-functions for any or all of these bound states. Do a nice graph and maybe compare them to the corresponding bound state from a infinite square well of the same width.)
*The way I envision this is that we work on this problem collaboratively here, freely sharing and comparing our ideas, questions and results. And, in addition, you hand in a nicely organized written solution. Not one or the other, but both. Does that make sense?
HW1 -part4. Hydrogen state related problems.
5. For an electron bound to a single proton (in what we call a hydrogen atom):
a) What is the energy of the electron ground state?
b) What is the kinetic energy of the electron ground state?
c) What is the potential energy of the electron ground state?
d) What is the ground state wave-function? Write this in the form \(e^{-r/b}\) with a normalization factor also written in terms of b. (Subsume hbar, m and e into the parameter b.) What are the units of the parameter b?
[Note: we are using b here because we will use a for the lattice constant in crystal lattices.]
e) In terms of x, y and z, what is r? Graph r as a function of x for y=z=0. Graph the ground state wave-function as a function of x for y=z=0.
f) What is the "size" of the ground state? Discuss here how one might quantify that.
6. a) How many first excited states are there for the hydrogen atom? How many linearly independent first excited states are there for the hydrogen atom?
b) The \(\psi_{n,l,m}\) first excited states that you learned about in physics 102 can be written as a linear combination of a state proportional to \(xe^{-b/2a}\) and 3 other states. What are the 3 other states? Which one is very different from the others even though it has the same energy?
[Extra credit: comment here on what symmetry leads to this extra degeneracy, and, also on what symmetry leads to the 3-fold degeneracy.]
c) Express the \(\psi_{n,l,m}\) first excited states in terms of your 4 states from part b.
7. Use 3 of your 4 first-excited states from problem 5 to construct sp2 first excited states. Visualize and discuss the essential nature of these sp2 states (and the leftover state). How are these important to life on earth?
8. Use all of your 4 first-excited states from problem 5 to construct sp3 first excited states. Visualize and discuss the essential nature of these states. How are these important to semiconductor physics and all solid state devices (phones, computers...)?
a) What is the energy of the electron ground state?
b) What is the kinetic energy of the electron ground state?
c) What is the potential energy of the electron ground state?
d) What is the ground state wave-function? Write this in the form \(e^{-r/b}\) with a normalization factor also written in terms of b. (Subsume hbar, m and e into the parameter b.) What are the units of the parameter b?
[Note: we are using b here because we will use a for the lattice constant in crystal lattices.]
e) In terms of x, y and z, what is r? Graph r as a function of x for y=z=0. Graph the ground state wave-function as a function of x for y=z=0.
f) What is the "size" of the ground state? Discuss here how one might quantify that.
6. a) How many first excited states are there for the hydrogen atom? How many linearly independent first excited states are there for the hydrogen atom?
b) The \(\psi_{n,l,m}\) first excited states that you learned about in physics 102 can be written as a linear combination of a state proportional to \(xe^{-b/2a}\) and 3 other states. What are the 3 other states? Which one is very different from the others even though it has the same energy?
[Extra credit: comment here on what symmetry leads to this extra degeneracy, and, also on what symmetry leads to the 3-fold degeneracy.]
c) Express the \(\psi_{n,l,m}\) first excited states in terms of your 4 states from part b.
7. Use 3 of your 4 first-excited states from problem 5 to construct sp2 first excited states. Visualize and discuss the essential nature of these sp2 states (and the leftover state). How are these important to life on earth?
8. Use all of your 4 first-excited states from problem 5 to construct sp3 first excited states. Visualize and discuss the essential nature of these states. How are these important to semiconductor physics and all solid state devices (phones, computers...)?
Thursday, March 24, 2016
HW 1 -part 1: Square well states and energies.
Please let me know of any typos no matter how small. Thanks. Also, question, comments and feedback are greatly appreciated! I may add more, but you can start this anytime.
Regarding units, in this class please use eV (for energy) and nm for length.
Semiconductor physics is based on quantum mechanics and understanding quantum states and energies. These problems are important to establish a foundation for this class.
If you run across any terms or concepts that you don't understand, and I expect that you will, please ask about them here. You get credit for questions. If you don't ask, I'll tend to assume that you understand everything, or that you haven't read the problems, or that you are reluctant for some reason to engage and interact. I feel that education is most effective when you do choose to engage and interact. I am not a fan of a "passive student" approach.
1. a) What width of infinite square well has a ground state energy of 10 eV?
b) For that square well, what is the numerical value of the wave-function in the center of the well?
c) If we center the well at x=0 then the wave function is A cos(kx). What is k? What are its units?
**Please post your thoughts, comments and answers to this question as a comment to this post as soon as you work on it. That is, post right away. Today. Now.
2. Consider a finite square well of the same width as you found for problem 1. Call this L. Let's center this well at x=0. Suppose the bottom of the well is at V(x) = -20 eV, and outside the well V(x)=0 eV.
a) what is the energy of the ground state more or less? (please post this as soon as you work on this. ASAP!)
[Let me suggest some notation here for square well problems including this one. Center the well at x =0. Inside the well write:
\(\psi_1 (x) = A \sqrt{\frac{2}{L}}cos (kx)\) where A and k are to be determined via bc's and wave equation.
outside right side write:
\(\psi_1 (x) = B \sqrt{\frac{2}{L}} e^{-(x-L/2)/b}\). L is the width of the well, right? b is a key parameter to be determined. B is a normalization parameter.
You can figure out b and k, and the state energy, without determining A and B. In this notation, by the way, A and B are unitless, and I think that A is something that can be compared to one to see how much the wave-function amplitude is diminished. Does that make sense?]
b) Sketch the ground state (gs) wave-function.
c) What is the numerical value of the wave-function in the center of the well? Is it smaller or larger than the gs wf of the infinite sq well of problem 1. Why? Discuss briefly. (Post your results and thoughts ASAP.)
d) How far outside the well does the wave-function extend? Comment below if you are not sure what this is asking. How might one quantify this? Speculate.
e).... what else should we explore for this well...? (Post you thoughts or ideas here.)
Note: It might make sense for you to skip problem 3 and go straight to problem 4. It is up to you. At any rate, don't get too bogged down on problem 3. If you have trouble with it you may want to skip it and go on to problem 4, which is a little more important.
3. Consider the limit of an infinitely narrow square well. That is, \( V(x) = -\alpha \delta(x)\) .
a) For what value of alpha is the energy of the ground state -10 eV? (What are the units of alpha?)
b) What is a characteristic length scale associated with this wave function? What is the "size" of the electron in this state? (To what region of space is it localized?)
c) Discuss how this wf relates to the gs wf of a finite square well.
d) What are the electon's KE and PE?
e) what else should we explore for this state?
**I'll make a separate post for question 4 so that we can work on it collaboratively there via comments on that post. Please go to that post to comment on problem #4.
4. Consider a well of the width we found for problem 1 of this homework but with a depth of 100 eV:
a) How many bound states does it have and what are their energies.
... (see HW1: part2 post)
Regarding units, in this class please use eV (for energy) and nm for length.
Semiconductor physics is based on quantum mechanics and understanding quantum states and energies. These problems are important to establish a foundation for this class.
If you run across any terms or concepts that you don't understand, and I expect that you will, please ask about them here. You get credit for questions. If you don't ask, I'll tend to assume that you understand everything, or that you haven't read the problems, or that you are reluctant for some reason to engage and interact. I feel that education is most effective when you do choose to engage and interact. I am not a fan of a "passive student" approach.
1. a) What width of infinite square well has a ground state energy of 10 eV?
b) For that square well, what is the numerical value of the wave-function in the center of the well?
c) If we center the well at x=0 then the wave function is A cos(kx). What is k? What are its units?
**Please post your thoughts, comments and answers to this question as a comment to this post as soon as you work on it. That is, post right away. Today. Now.
2. Consider a finite square well of the same width as you found for problem 1. Call this L. Let's center this well at x=0. Suppose the bottom of the well is at V(x) = -20 eV, and outside the well V(x)=0 eV.
a) what is the energy of the ground state more or less? (please post this as soon as you work on this. ASAP!)
[Let me suggest some notation here for square well problems including this one. Center the well at x =0. Inside the well write:
\(\psi_1 (x) = A \sqrt{\frac{2}{L}}cos (kx)\) where A and k are to be determined via bc's and wave equation.
outside right side write:
\(\psi_1 (x) = B \sqrt{\frac{2}{L}} e^{-(x-L/2)/b}\). L is the width of the well, right? b is a key parameter to be determined. B is a normalization parameter.
You can figure out b and k, and the state energy, without determining A and B. In this notation, by the way, A and B are unitless, and I think that A is something that can be compared to one to see how much the wave-function amplitude is diminished. Does that make sense?]
b) Sketch the ground state (gs) wave-function.
c) What is the numerical value of the wave-function in the center of the well? Is it smaller or larger than the gs wf of the infinite sq well of problem 1. Why? Discuss briefly. (Post your results and thoughts ASAP.)
d) How far outside the well does the wave-function extend? Comment below if you are not sure what this is asking. How might one quantify this? Speculate.
e).... what else should we explore for this well...? (Post you thoughts or ideas here.)
Note: It might make sense for you to skip problem 3 and go straight to problem 4. It is up to you. At any rate, don't get too bogged down on problem 3. If you have trouble with it you may want to skip it and go on to problem 4, which is a little more important.
3. Consider the limit of an infinitely narrow square well. That is, \( V(x) = -\alpha \delta(x)\) .
a) For what value of alpha is the energy of the ground state -10 eV? (What are the units of alpha?)
b) What is a characteristic length scale associated with this wave function? What is the "size" of the electron in this state? (To what region of space is it localized?)
c) Discuss how this wf relates to the gs wf of a finite square well.
d) What are the electon's KE and PE?
e) what else should we explore for this state?
**I'll make a separate post for question 4 so that we can work on it collaboratively there via comments on that post. Please go to that post to comment on problem #4.
4. Consider a well of the width we found for problem 1 of this homework but with a depth of 100 eV:
a) How many bound states does it have and what are their energies.
... (see HW1: part2 post)
Physics 156 Introductory post
I would very much recommend that you start HW 1 -part 1 as soon as you have time. I believe that it would be very valuable to do it over several days so that you have time to mull it over and really think about and absorb the different aspects of quantum that the assignment covers.
HW1 -part 2 will cover propagating states and excited states. (high momentum states).
A book that might be relevant to this class, though not this first assignment, is Streetman, Solid State Electronic Devices. (any)
HW1 -part 2 will cover propagating states and excited states. (high momentum states).
A book that might be relevant to this class, though not this first assignment, is Streetman, Solid State Electronic Devices. (any)
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