As Taught In:
2020/2021
Level:
Undergraduate
Learning Resource Types:
=> Problem Sets
=> Notes
=> Reading Resources
Course Overview:
Together with 105 – Intermediate to Quantum Mechanics, this is a continuation of beginning Quantum Mechanics. It introduces some of the important model systems studied in contemporary physics, including two-dimensional electron systems, the fine structure of hydrogen, lasers, and particle scattering.
Prerequisites:
Physics 105 – Intermediate to Quantum Mechanics
Course Goal:
By the end of this course, learner will be able to interpret and analyze a wide range of quantum mechanical systems using both exact analytic techniques and various approximation methods. This course is a continuation of Physics 105 – Intermediate to Quantum Mechanics, and will introduce some of the important model systems as define in section “Course Overview”
Textbook:
Griffiths, David J. and Darrell F. Schroeter. Introduction to Quantum Mechanics. Cambridge, United Kingdom: Cambridge University Press, 2018. ISBN: 9781107189638.
Cohen-Tannoudji, Claude, et. al. Quantum Mechanics, Vol. 2. Wiley, 1991. ISBN: 9780471164357. (Highly recommended) Vol.1. is also good.
Merzbacher, Eugen. Quantum Mechanics. Wiley, 2005. ISBN: 9788126533176. (Highly recommended)
Shankar, Ramamurti. Principles of Quantum Mechanics. Plenum Press, 2011. ISBN: 9780306447907. (Recommended)
Sakurai, J. J. Modern Quantum Mechanics. Cambridge University Press, 2017. ISBN: 9781108422413.
Schumacher, Benjamin, and Michael D. Westmoreland. Quantum Processes, Systems, and Information. Cambridge University Press, 2010. ISBN: 9780521875349.
Feynman, Richard. P. Feynman Lectures On Physics. Vol. 3. Basic Books, 2011. ISBN: 9780201021158.
Ohanian, Hans. Principles of Quantum Mechanics. Prentice Hall, 1989. ISBN: 9780137127955.
There are three major parts:
Part 1: Time Independent Perturbation Theory and WKB Approximation
1. Time Independent Perturbation Theory
2. Degenerate Perturbation Theory
3. Hydrogen Atom Fine Structure
4. Zeeman Effect and Introduction to the Semiclassical Approximation
5. The Semiclassical WKB Approximation
6. Airy Functions and Connection Formulae
7. Time Dependent Perturbation Theory
Part 2: Time Dependent Perturbation Theory and Adiabatic Approximation
8. Fermi’s Golden Rule
9. Fermi’s Golden Rule for Harmonic Transitions
10. Hydrogen Ionization – Lights and Atoms
11. Charged Particles in Electromagnetic Fields
12. Adiabatic Approximation
13. Adiabatic Approximation: Berry’s Phase
14. Adiabatic Approximation: Molecules
Part 3: Scattering and Identical Particles
15. Scattering
16. Identical Particles
References:
[G] = Griffiths, David J. and Darrell F. Schroeter. Introduction to Quantum Mechanics. Cambridge, United Kingdom: Cambridge University Press, 2018. ISBN: 9781107189638.
[CT] = Cohen-Tannoudji, Claude, et. al. Quantum Mechanics, Vol. 2. Wiley, 1991. ISBN: 9780471164357.
[Sh] = Shankar, Ramamurti. Principles of Quantum Mechanics. Plenum Press, 2011. ISBN: 9780306447907.
[Sa] = Sakurai, J. J. Modern Quantum Mechanics. Cambridge University Press, 2017. ISBN: 9781108422413.
Readings:
Week 1 >> [G] Chapter 6, [CT] Chapter XI, [Sh] Chapter 17, [Sa] Sections 5.1-5.3
Week 2 >> [G] Chapter 6, [CT] Chapter XI & XII, [Sh] Chapter 17
Week 3 >> [G] Sections 6.3-6.5, [CT] Chapter XII, [Sh] Chapter 17
Week 4 >> [G] Chapter 8 on WKB approximation, [Sh] Section 16.2
Week 5 >> [G] Chapter 9, [CT] Chapter XIII, [Sh] Chapter 18
Week 8 >> [G] Sections 9.2 – 9.3 for interactions of atoms with light. Chapter 10 for the adiabatic approximation
Week 9 >> [CT] Chapter VI Complement E, which has a nice treatment of Landau levels. [Sh] treatment of Landau Levels begins from page 587, [Sa] pg. 130-139
Week 11 >> [G] Chapter 11, [CT] Chapter VII, [Sh] Chapter 19
Week 13 >> [G] Chapter 5, [CT] Chapter XIV, [Sh] Chapter 10
As I mentioned in other course that I have no fans on the paper exams. However, the condition made us do a paper exam. Therefore, this is an open-book exam that needs to be solved within the time frame by the end of each quarter in the course.
In this course there will be ten problem sets for this last course of beginning Quantum Mechanics. Problem sets will be posted on the course website at least one week before they are due. Solutions will be available on the course website the day after the problem set is due.
For practical, not punitive, reasons, late homework will not be graded under any circumstances. This is especially important if you think you might want to change from listener to credit status. For conflicts that are known in advance, such as religious holidays or travel, problem sets should be turned in before the deadline. If you are away from on-site, you may submit your problem set digitally, as long as you alert the Teaching Assistant ahead of time. To allow for unforeseen circumstances such as illness or emergencies, one problem set will be dropped from the average work, either an omitted set or the one with the lowest score. We strongly recommend that you nevertheless turn in all of the prob-sets, and that even if you are unable to turn one in, that you work through the problems on your own anyway.
Problem sets are a very important part of this class. We believe that sitting down yourself and trying to reason your way through a problem not only helps you learn the material deeply, but also develops analytical tools fundamental to a successful career in science. We recognize that learners also learn a great deal from talking to and working with each other. We therefore encourage each learner to make his/her own attempt on every problem and then, having done so, to discuss the problems with one another and collaborate on understanding them more fully. After you have understood the problem, it is essential for your understanding to write up the solution completely by yourself. It is a breach of academic integrity to copy any solution from another learner or from previous years solutions. Your solutions should be logical, complete and legible. If you cannot present a solution clearly, it is likely that you do not understand it adequately. The graders are instructed not to give credit for unclear or illegible solutions.
in Summary:
Everyone in this course will be expected to research, write and “publish” a short paper on a topic related to the content of Physics 105 Intermediate to Quantum Mechanics or Physics 106 Advanced to Quantum Mechanics. The paper can explain a physical effect or further explicate ideas or problems covered in the courses. It can be based on the learner’s own calculations and/or library research. The paper should be written in the style and format of a brief journal article and should aim at an audience of Physics 106 learners.
Because Physics 106 is a Communication Intensive in the Major, in order to pass learner must do writing, editing, revising and “publishing” skills are an integral part of the project. Each of you will ask another student to edit your draft and will then prepare a final draft on the basis of the suggestions of your “peer editor.” We will supply templates for the RevTeX version of LaTeX (used by the Journal Physical Review) so that you can prepare your paper in a finished, publishable, form. In there is sufficient demand, we can also arrange a LaTeX tutorial.
You will submit your first draft marked up with editorial comments by your peer editor. This first draft will then be critiqued by a “writing assistant” (see below) and returned to you. Two weeks after the first draft is due, you will submit your final draft. Your papers will be graded on the intellectual quality of your work, the effectiveness of your presentation and the success of your prose style. A part of your grade will also be determined by how carefully and constructively you edited the draft of the paper for which you were the peer editor. The grade you earn for your paper will count 25% towards your final grade.
A practicing physicist writing a research paper often asks a few colleagues to comment on a first draft. The final draft is then reviewed anonymously by one or several peers before it is accepted by a journal like the Physical Review. The goal of this informal and formal peer review process is to push authors to write papers which successfully communicate ideas among a community of peers. Your goal is to write a paper which presents a phenomenon or problem in quantum physics in a way which communicates your ideas clearly and effectively to your fellow Physics 106 learners, namely to your peers. If your peers cannot understand what you write, you have not succeeded. Note that writing for your peers is a much higher standard than writing for the faculty. Presenting a topic sufficiently clearly and logically that one of your peers new to this topic can learn about it requires clarity of thought and depth of understanding. These are the prerequisites for an effective written (or, for that matter, verbal) presentation.
We have “writing assistants” who can help you with writing, editing and preparing the paper. Each of you will be contacted by email by one of the writing assistants during the 8th week of the course. You should arrange to meet soon thereafter, and should seek their assistance from then on as you need it. They will critique the proposal and outline for your paper, and will also critique the first draft which you submit after it has been peer edited. In between, you may also ask them to help you with parts of your paper as you write them. Think of your writing assistant as a coach. They are there to help you, and are good at it.
By the time you turn in your final paper, it will have been edited by one of your peers and you will also have had time to implement the suggestions of one of the writing assistants. Past Physics 106 learners have found that their papers improve enormously through this process. Based on experience from previous years, by the time you turn in your finished paper, very many of you will have produced an account of a piece of physics written to a very high standard. It would be a shame if these papers were not “published.” We shall have as our goal the “publication” of a journal consisting of all your papers. Note that for publication it is important that you submit your paper electronically, using the LaTeX template provided. Subject to these caveats, we hope to produce a compilation of all of your papers. We will circulate this “journal” to all of you, so that you can, in the end, read the work of all your peers.
The aim of this project is to give a clear and pedagogical presentation of a “problem” or “phenomenon” in quantum mechanics.
- A “problem” could be similar to but more elaborate than the type of problems that appear on problem sets. For example, squeezed states were introduced briefly in the context of the harmonic oscillator in Physics 105 – Intermediate to Quantum Mechanics. A student might delve deeper into the squeezed state formalism, describe the properties of squeezed states, explain the types of problems where they are useful, and give some examples of their applications. Such a paper would resemble a short chapter in some hypothetical textbook for Physics 105. The principal references for a paper like this could be existing quantum mechanics texts and the references to the original literature to be found in them.
- A paper focused on a “phenomenon” would introduce the phenomenon and explain its origins in terms of the concepts and language of Physics 106. For example, in studying the properties of atoms, we will briefly mention phenomena such as line-width, Thomas precession, and the Lamb shift. To write a paper about atomic line-widths, for example, a student might find out what some of the dominant types of line broadening are, when and how one can reduce them, and might talk about experiments that have managed to do so. Once again the principal references would likely be texts, perhaps modern physics texts in this case, histories of quantum physics, and the original literature.
Papers on “problems” might be based at least in part on your own calculations. Papers on “phenomena” might involve some library research. In either case reference must be given for any material taken from other sources. Do not plagiarize. Anyone who contemplates borrowing material directly from mainstream texts should consider how difficult it is to find a text that presents quantum physics at the level appropriate to Physics 106.
We encourage learners to write papers which expand upon a problem or phenomenon which was already introduced in either Physics 105 or Physics 106 lectures. If you do this, you should begin at the level of whatever we have already covered and then go farther. Students may also choose topics which have not appeared at all in class, but whose quantum mechanical explanation can be understood based upon what we have learned in Physics 105 and Physics 106.
Please do not try to choose subjects which are obscure, difficult, or controversial. Misguided attempts like this to gain the respect of the faculty inevitably have the opposite effect. There are plenty of deep, interesting, and challenging subjects in the mainstream of quantum mechanics.
Papers can range between 3.5 and 7 pages in length (in the RevTeX final, two-column format) including references and figures. These limits are firm. Students are encouraged to use equations and figures to aid their presentation, much as they are used in articles and textbooks.
Ingredients of a good term paper
Ideally, a term paper should contain some calculation and some context (historical, mathematical, physical, engineering, etc.). The sample paper on reflectionless potentials does a good job of showing what level of calculations is right: enough detail for an Physics 106 learner to follow, but not an excessive amount. However, there are a few ways that your own papers should differ. First, this paper shows more originality than you need to: it gives a solution to a problem that has not previously been published (even if it had been “folklore” before), which is not reasonable to ask of you. On the other hand, it would be better if it gave more of a connection to a real-world scenario. Given the topic, this would be difficult, but the discussion section still gives an idea of the type of material that is good to include.
You should use the first part of the term to consider possible topics and to choose a peer editor. Your peer editor must be an Physics 106 learners, and must be someone whose own Physics 106 paper topic is unrelated to yours. A list of suggested topics is given below, but you are free to choose topics not on this list upon first obtaining Educator’s team approval.
Your proposal is due on in your Friday section during week 8 and must be submitted digitally by 12:00 noon of local time. This must consist of: a title, a one paragraph description of what you plan to write about, an outline of your proposed paper, a list of several references you plan to use, the name of your peer editor, and your name and email address.
You will then be contacted by one of the writing assistants. You and your peer editor will have the same writing assistant. They may either accept your proposal, or request that you revise it in response to their suggestions. You and your peer editor must arrange to meet with your writing assistant as soon as possible (even if they accept your proposal).
There is a peer editing session your Friday section during week 10. At that time, you and your peer reviewer will exchange drafts and make constructive, written comments on each other’s work.
The quality of the term papers improve dramatically from peer comments, so it is recommended that you implement their suggestions before submitting your first draft. To demonstrate that you participated in the peer editing session, you should submit a copy of your peer editing comments for grading.
Your first draft is due on Friday in week 11, when you will submit a pdf version electronically. Each of you, together with your peer editor, should then meet with your writing assistant by by the end of week 12 in order to obtain their comments on your first draft. You will get your first drafts back when you meet with your writing assistant.
A final copy of your final, polished paper, will be two-column options is due by Friday of week 13, when a pdf version should be submitted digitally. Think of this as submitting your paper to the Physics 106 Paper Review, and our “publisher” will assemble the pdf files into a bound document.