PHYS 231 Fall 2020
Methods of Theoretical Physics (ARCHIVED)
Methods of Theoretical Physics (ARCHIVED)
Prof. Flip Tanedo (flip.tanedo@ucr.edu)
TA: Ian Chaffey (ichaf001@ucr.edu)
This is a crash course on the mathematical methods to succeed in the first year graduate curriculum. Our goal is to solve differential equations using Green's functions. We will use techniques from linear algebra and complex analysis. If time permits, we may explore other topics toward the end of the course such as statistical methods for physics and astronomy.
Remote Course Format: for Fall 2020 this class will be offered remotely. We will guide our way through readings and short, pre-recorded videos. Your primary homework will be a weekly explainer video and peer reviews for those videos.
Our nominal scheduled time slots are: MWF 10:00 - 10:50am (California time) and M 3-3:50pm (California time). We will not use this for lectures, but will occasionally schedule interviews during this period.
Syllabus; please review (I reserve the right to update policies as needed)
Course Notes: updated as we go, your feedback and corrections are welcome.
Grading rubric for explainer videos
Internal Course Page (registered students only; for sharing explainer videos)
This course will be held remotely and largely asynchronously. We will communicate primarily through our Slack channel. We will coordinate meeting times for one-on-one interviews in the first week class.
There is no required textbook, but you are strongly encouraged to have access to at least one general "math methods for physicists" reference.
I'll be updating a set of course notes; feedback and corrections are welcome. You may also refer to the notes from 2018 and 2017.
The closest "textbook" to our course is Mike Godfrey's P30672 course in Manchester; there's a nice set of notes here. I'm not following them, but they do seem close to my philosophy for this class.
I particularly like Mathematics for Physics & Physicists by Appel, but we will not necessarily follow it closely.
Mathematics of Classical and Quantum Physics, Byron and Fuller. Inexpensive gem as a Dover edition.
Mathematical Physics: a Modern Introduction to its Foundations, Sadri Hassani; digital version free from UCR Library.
Mathematical Methods of Physics, Mathews and Walker
Mathematical Methods in the Physical Sciences, Mary Boas; primarily undergraduate level, but is an excellent background reference for most of the topics in this class.
Feel free to look for some that you can access electronically from the UCR Library.
Course notes: An ongoing set of course notes will be typed up as we go.
You may need to use your @ucr email address to access certain course materials, a regular @gmail address will not work. Other materials may be posted only in iLearn. Here are instructions for using the UCR VPN if you want to avoid using the UCR login each time you access these materials.
For this course, you will need to be able to record 5-10 minute videos of yourself explaining the solutions to homework problems. There are many ways to do this, check out the UCR Keep Learning website for suggestions. Your videos do not need to be polished: you need to be effective, not flashy.
I encourage you to have a recording that shows your face while talking if possible. This will help us build familiarity with one another. At the very minimum, your videos should have the sound of your own voice.
In the second half of the course you will prepare a short written document explaining how to solve for the Green's function of a harmonic oscillator. You are strongly encouraged to use LaTeX.
We will use Zoom to hold one-on-one interviews and Slack for broader communication. Registered students will receive email invitations.
This is an unusual year to be a first-year graduate student. Our goal with this course is to adapt in a meaningful way to make this experience valuable. Our guiding principles are:
Your time and attention are precious. Remote learning can quickly cause Zoom fatigue. We want to allow as much flexibility in your lives.
Your words are more important than mine. Rather than traditional lecture, it's better for us to find more meaningful ways to engage with the material and each other. This will be somewhat of a "flipped classroom" where we focus on you solving problems rather than hearing me tell you about solving problems.
Your cohort is valuable. Your grad school colleagues are your most valuable allies. Even though we may not be in the same place, let's learn to find effective ways to collaborate with one another. Respect one another.
This is a "support" class. There is no comprehensive exam test for this course. The main purpose of this course is to make sure you have the mathematical tools to succeed in your grad classes this year. This class is not trying to "weed out" any students or place undue burden on your attention.
Communication is key. As young scientists, you will be judged as much on your ability to communicate your science verbally as your ability to "do" your science. You present your homework as videos because this is training for every oral exam, conference talk, and interview that you have ahead of you.
You get what you put in. You must work through problems to get anything out of this course. You don't have to do many and it doesn't have to take much time, but you will get nothing from just "watching" this course.
The purpose of these assessments is for you to be able to engage meaningfully with the material while learning from each other. The format and choice of problems is because in all of my hindsight, this is exactly how I wish I were trained in my first year of grad school to make me a better physicist.
I reserve the right to modify the homework load and type as necessary during the course.
Your grade will be based on:
Explainer videos (35%): Each week you will prepare and upload one video explaining how to do one of the homework problems.
Peer review (35%): Each week you will review five videos by your peers. You will grade them based on a rubric and you will provide constructive feedback.
Interviews (10%): Each week you will have a 10 minute interview with either Flip or Ian to discuss your homework and the course. These interviews will help validate the peer review grades and give us feedback on what topics to review in class. See this form for an example for how we will be assessing the interviews.
Surveys (10%): Each week you will complete a short survey with review questions and requests for metacognitive feedback.
Essay (10%): In the second half of the course, you will have a one-time written assignment to prepare a "how to" guide on solving the harmonic oscillator using the techniques in the course.
Slack Engagement (10%): You are expected to contribute to the discussion on Slack by either asking questions, answering your peer's questions, sharing useful resources, and otherwise contribution to the class culture.
There will be no final exam, you do not need to arrange to be present during the final exam slot for this course.
See the keeplearning.ucr page for advice on recording and uploading your videos. You will upload your video to your own storage space (e.g. the unlimited storage volume through your @ucr access to Google Drive) and then submit it by sending the link through a submission form.
Please keep your videos accessible until the end of the term (3rd week of December).
Please make sure that your uploaded video is viewable by other members of the class.
On Google Drive (web interface): right-click on the file and click on "Share." There are a few options:
Restricted: only people listed can access the video. Please add flip.tanedo@ucr.edu, ichaf001@ucr.edu, and the email addresses of your classmates (see the internal page).
Rmail: automatically restricts to anyone with an @ucr email address.
Anyone with the link: least restrictive, public access. I don't recommend this, but you can do this if you'd like to share your videos with family and friends outside of UCR.
Welcome to the course. Testing technology.
Lecture 01: Introduction to Physics 231. Tip: if you watch the video at 2x speed it takes half the time. (Video linked below.)
Week 0 survey: welcome! Due: Wed, 10/7. Please fill out this introductory survey to help the teaching class calibrate this year's course.
Introduction video. Due: Wed, 10/7. Please prepare a video introducing yourself to the rest of the class. Your video should be 2-5 minutes long. My suggested topics to cover are: your name (as you would like to be referred), where you're from, activities that you enjoy, research interests, favorite book/television show/podcast, etc. Here's a sample video from Flip.
Dimensional analysis, linear algebra review, "indexology."
Lec 1: Dimensional Analysis
Lec 2: Examples of Dimensional Analysis. Remark: in the treatment of error, epsilon is a percent error so that the time to fall t is t0 times (1+epsilon). Dimensional analysis tells us that epsilon = h/R. Thanks to Alec P. for catching this.
Lec 3: Linear Algebra Review: indices
Lec 4: Function spaces, derivatives as linear transformations
Please feel free to read as much or as little of the following references as you feel appropriate for what you'd like to get out of this course.
"Dimensional analysis, falling bodies, and the fine art of not solving differential equations," Craig Bohren.American Journal of Physics 72, 534 (2004); https://doi.org/10.1119/1.1574042 (access through UCR VPN)This article really captures the spirit of this course and is a non-trivial demonstration of the power of dimensional analysis. It gives us a reason to pause and think about what it means for our idealized models of nature to be reasonable approximations to the complex reality around us.
"Natural Units and the Scales of Fundamental Physics," Robert Jaffe, Supplementary Notes for MIT’s Quantum Theory Sequence, Feb 2017. Robert Jaffe's notes have plenty of examples of dimensional analysis as well as a thorough introduction to natural units.
"Introduction to Tensor Calculus," Kees Dullemond & Kasper Peeters (booklet) 2010.This is a reasonable introduction to the "indexology" that we use in physics. The booklet is much more rigorous than we will be. It may help motivate the types of rules that we will build into our index notation. If you enjoy this reference, you can find more in the first few chapters of most general relativity textbooks, for example Sean Carrol's Spacetime and Geometry.
"Tensors: A guide for undergraduate students," Franco Battaglia, American Journal of Physics 81, 498 (2013); https://doi.org/10.1119/1.4802811 (accessible through UCR VPN)This is an excellent and readable introduction to to tensors. In the last half of the article (beyond our scope) one starts to get a taste of how linear algebra is a local approximation to calculus on curved spaces.
"Section 11: The Method of Similarity," in Mathematical Methods of Classical Mechanics by V.I. Arnold. This is a delightful book about differential geometry that is disguised about a book about mechanics. In section 11 Arnold describes how to use scaling relations to relate relate different orbits of a central potential (we mention this example in our class notes). Much more fun are the two problems at the end of the section:
Differential equations as linear algebra. Function spaces.
Lec 1: Why are all of our favorite differential equations first or second order?
Lec 2: Row vectors, dual vectors, covariant vectors, one-forms. They're all the same thing.
Lec 3: Bra-Ket notation, eigenvectors and eigenvalues.
Lec 4: The instructive example of the electrostatic potential of a charged cat.
Lec 5: Metrics convert vectors to dual vectors.
Read as much or as little as you feel necessary for your our goals in this course.
See the tensor references from Week 1.
See Section 2 and Section 3.1-3.2 of the Manchester notes.
Section 4.1 of Stone & Goldbart distinguishes between formal and concrete differential operators. For our purposes, a formal differential operator is what one would typically call "a differential operator," examples include the d'Alembertian or the Laplacian. A concrete differential operator comes with a specification of the domain and the boundary conditions.
Chapter 4.1-4.8 of Byron & Fuller is a nice summary of finite dimensional vector spaces. Chapter 5 introduces function spaces.
Differential equations as linear algebra. Eigenfunctions, Green's functions.
I forgot to upload lectures 6 and 7 from week 2! Apologies.
Week 2, Lec 6: Indices, the adjoint
Week 2, Lec 7: Green's function problem in function space
This week's videos (ongoing list):
Week 3, Lec 1: dual vectors in function space, completeness in function space
Week 3, Lec 2: completeness and orthonormality in function space
Week 3, Lec 3: completeness and Green's functions
Week 3, Lec 4: why the series solution is not the whole story
Week 3, Lec 5: the patching solution
Week 3, Lec 6: examples: solving a simple Green's function with the series solution and the patching solution
Section 4.1 of the Manchester notes.
In our lectures we discuss why there there are really only a few differential operators that always keep showing up in physics. Part of this has to do with the symmetries that our physical models satisfy, and part of it has to do with the idea that terms with more derivatives are typically small in the limit where our model is valid. This falls under the idea of an "effective theory." This year there's a virtual seminar series about effective field theories (all talks are recorded) that I recommend for those who are curious.
Extra: "Dimensional Analysis in Field Theory," Stevenson, Annals of Physics 132, 383 (1981). A non-trivial example of dimensional analysis.
Homework 1. Your first assignment. You will be required to create a video for one of these problems. These are posted on the internal course page.
Week 3 Survey. The link on our internal course page.
Complex analysis review. Complex functions. Integration.
Lec 1: Teaser---why are we studying complex numbers? A "bird's eye view."
Lec 2: Complex numbers and complex functions
Lec 3: Analytic = complex differentiable = independent of z*
Lec 4: Complex functions as transformations of the complex plane; don't integrate across branch cuts
Lec 5: Integration in the complex plane
Lec 6: Cauchy Theorem: integrals of analytic functions are boring
Lec 7: Optional: a hint of Stokes' theorem
Lec 8: Cauchy Integral Formula
Lec 9: Introduction to the Residue Theorem
Bonus Lec: Unsolicited grad school advice / Week 4 pep talk.
Please review complex analysis. Our goal is to be able to perform complex contour integrals and to use the residue theorem.
Byron & Fuller, Chapter 6
Boas, Chapter 14 (friendly introduction)
Matthews & Walker, 3-3 and Appendix A
Stone & Goldbart, Ch. 17 (succinct with lots of discussions of advanced topics; suggested for those who are already familiar with complex analysis)
Appel, Chapter 4.1-4.6
Cahill, Chapter 5.1-5.14
Peter Oliver's notes (UMN) on complex analysis and conformal mapping
Extra: "Why i?" Baylis, Huschilt, and Wei. American Journal of Physics 60, 788 (1992); https://doi.org/10.1119/1.17060While not quite the direction we're going, students often bring up quaternions and the Pauli matrices when we discuss complex numbers in physics. This article is a nice introduction.
Extra: Alexandre Eremenko's notes on "Why Airplanes Fly and Ships Sail," an introduction to conformal mapping and fluid dynamics
Some background information in an SPS article by Dwight Neuenschwander, "How Airplanes Fly: Lift and Circulation"
All assignments can be accessed from our internal page. (This is to protect student information in accordance with FERPA and to keep our shared work "in house.")
Peer Reviews
Survey
Interviews next week
Complex analysis: contour integrals.
We will have an office hour on Wednesday, Nov 10.
Lec 1: Residue theorem more carefully
Lec 2: non-simple poles, a cautionary tale
Lec 3: using complex analysis for real integrals
Lec 4 (extra topic): motivation of analytic continuation
Lec 5: the main example; follow this carefully!
Lec 6 (extra topic): a fancy example with a branch cut
Same as the Week 4 references for complex analysis.
While this has nothing to do with our course material, every time I teach this class I am reminded of something called "analytic continuation into superspace" where the idea (or at least the spirit) of analytic continuation is applied to a particular complexification of spacetime called superspace.
On Wednesday we will have our first round of interviews. The schedule is on our internal page. See this form to see how we will be assessing the interviews. (Form is subject to change during the course of the term. )
No assignments for week 5. To be posted.
Green's functions: putting it all together; the harmonic oscillator.
Note: from week 7 onward we shifted to in-real-time lectures with recordings. Because these recordings include student questions, we are no longer posting them publicly. Class members may access the internal site.
You may follow what we talked about through our Course Notes.
Homework #4 (last one!) Due Friday, Dec 4.
Note: this one is a bit longer than usual, but it shouldn't be much harder. You can use up to 30 minutes to present your solution, but don't feel pressured to use all of this time. Make sure the presentation explains the problem and shows your peers how to think about the problem.
You only have to do one of the two problems, though they're a bit longer than usual. Your assignments are below.
Essay, Due Friday, Dec 11.
Prepare a set of pedagogical notes written for yourself (but submitted for Flip and Ian to review) on "how to the harmonic oscillator" using Green's functions. The notes should cover the following:
A brief reminder of what a Green's function is and how you use a Green's function to solve an inhomogeneous, linear differential equation.
A step-by-step guide on solving the Green's function for the 1D harmonic oscillator using Fourier transforms. Be sure to explain the prescription for adding a small imaginary piece to the poles of the Fourier integrand and how this relates to causality. Explain how to perform the Fourier integral using contour integration techniques.
Comment on how this generalizes to the wave equation in spacetime; you can refer to some of the physical lessons from Homework #4, but you don't have to go into gory mathematical detail. If you want to be very fancy, you can write about how this all looks in cylindrical or spherical coordinates (it'll come in handy) where Bessel functions and spherical harmonics appear.
Your goal is to have a set of notes of perhaps a few pages long that you can refer to whenever you need to remind yourself of the key ideas in this course.
I suggest using LaTeX to type up your notes. This is the standard for writing papers in physics and astronomy (with the strange exception of condensed matter experiment).
If you are new to LaTeX, you can use the default templates in Overleaf.
I have a LaTeX template that I use for my papers (Overleaf, GitHub). It's a little idiosyncratic.
The LaTeX files for our course notes are on GitHub.
The following advice from Andy Buckley may be helpful.
Peer Reviews, Due Wed, Nov 25. To be assigned.
Damped harmonic oscillator, the wave equation as a harmonic oscillator in spacetime. You may follow what we talked about through our Course Notes.
Same as Week 7.
The material for Weeks 9 - 10 are for culture. We'll be discussing ideas related to Green's functions in physics, but we will sample from a few different fields including quantum mechanics and probability theory. The topics are non-examinable and there are no further homework assignments. You may follow what we talked about through our Course Notes.
Same as Week 7.
Extra topics: the Gaussian integral, correlation functions, and the relation to Green's functions. Introduction to Feynman diagrams as a perturbation expansion for non-linear differential equations. AMA about graduate school, finding an adviser, what's next in your academic career. You may follow what we talked about through our Course Notes.
Turn in final explainer video and course essay. No peer review this week
Assignments and video materials will be linked from this page. (We may post review assignments on a separate, non-public page.) Interviews will be arranged by e-mail.
All assignments are to be submitted using Google Forms. Please use your UCR account to upload materials on a storage space that our class can access.
The primary way of asking questions, commenting, and engaging with the course is through our Slack workspace. Please use the appropriate week's channel for physics questions. Be sure to tag Flip or Ian if you want a specific response.
You can also contact Flip and Ian by e-mail, please use: [P231] in the subject line so that the message is not filtered out. Please expect a minimum 2 business day turnaround time for e-mail messages.
We abide by the UCR Keep Teaching Privacy and Security guidelines. Because lectures will be asynchronous and are recorded without an audience, there is no risk to student privacy when these lectures are made available. All student-generated recordings (explainer videos) will be hosted on the student's own web space with links only shared internally with members of the class. Students have full control of removing these files after the course.
Students are not allowed to download other students' videos without their permission. The instructor grants students the right to download any of the instructor's videos for their own instructional use. In order to maintain a strict separation of private information (including course participation), we have a separate internal course webpage where students may access one another's work for peer review.
Each year this course is slightly different based on the feedback of the previous years' students and the discretion of the instructor. The number of blatant errors should be decreasing (mostly monotonically) with time. The number of subtle errors is also decreasing adiabatically.