• I’ve added notes to your Collaborative Problem Set Solutions for EM PS#6, LA PS#6, QM PS#3; please recall that you need to be logged in to access these.
• Please compare the solutions in your Problem Set Notebooks to these solutions, updating or correcting as you need to. If you disagree or would like further clarification, please send me an email.

For Week 18, please read the following:

• EM: Griffiths 4.4, 5.1, 5.2
  
  • Reminder: video lectures here.
• LA: Lay 6.1-6.3 (see this post)
  
• QM: Townsend Ch. 4
  • Reminder: some related video lectures here; particularly Quantum Mechanics Concepts 6: Hamiltonian, Energy, and Time Dependence offers material closely connected to this week’s material.
• P/S: NMR; Fourier Methods.

• Remember to use nb.mit.edu (info on how to access here) or check with faculty for alternatives. Complete first pass of reading and enter annotations (questions, responses, requests for class time) by Tue. 9am and update through week as need.

This week’s schedule.

• Mon. Feb. 27:
  • 9:00 – 11:00: Quiz/Workshop
  • 12:30 – 1:30: EM (bring Griffiths)
  • 1:45 – 2:45: LA/QM (bring Townsend)
• Tue. Feb. 28:
  • 9:00 – 11:00: LA/QM (bring Townsend)
  • 12:30 – 2:30: LA/QM (bring Townsend)
• Wed. Mar. 1: MEET IN LAB 2 2238
  • 9:00 – 1:00: EM Lab Activity & Lecture (bring Griffiths)
• Thu. Mar. 2:
  • 9:00 – 11:00: EM (bring Griffiths)
  • 12:30 – 2:00: LA/QM (bring Townsend)
  • 2:00 – 3:00 EM & LA/QM Workshop

• PS#6/#3 Solutions Assignments can be accessed here.
• Note that you will need to be logged in to our program web-site to access the page linked above. If not logged-in, you can do so via this link.
• Note that many problems are assigned to more than one person. People assigned to the same problem are welcome to work together and submit one solution, or work separately and submit two. For EM, this is mostly just for back-up; as we’ve seen in QM, there are often multiple ways to approach these problems, so I thought alternative solution paths might be helpful.

Problem Sets to be completed in your corresponding PSNs by noon Sun. Feb. 26.

• EM PS#6: Griffiths: 4.11&4.17, 4.12, 4.15, 4.16(a), 4.20, 4.21, 4.33, 4.34, 4.35
• LA PS#6: Lay 5.5: 4&6, 8&12, 23, 24
• QM PS#3: Townsend: 3.8, 3.13, 3.15, 3.16, 3.17, 3.18, 3.20, 3.22, 3.24, 3.25

• EM notes: For 4.12, note that since the polarization is uniform, the P and dot product can be pulled out in front of the integral; the integral is now a modified version of an integral you carried out in the collected set of problem 2.7, problem 2.8, Example 2.3 and problem 2.12. For 4.34, you’ll get to show that the information in the problem statement means that the dielectric constant is given by 1+ x/d. For 4.35, you’ll get to use the Dirac delta function.
• LA notes: Lay uses the overbar to mean complex conjugate (recall that Townsend uses *), such that A = A*. For 4 and 6, follow the approach outlined in Example 2. For 8 and 12, use Example 6 (as directed). Problems 23 and 24 are related to what you did for the last problem on LA/QM Exam 1; some of you might be interested in seeing what happens if you replace the condition that A is an n x n real matrix with the property A^T = A with the condition that A is an n x n complex matrix with the property that A^T = A (in other words, A is Hermitian), in which case this might become an alternative proof that the eigenvalues of a Hermitian matrix operator are real.
• QM notes: Problem 3.8 should be straightforward. For 3.24, choose the simplest N (e.g. with phase = 0). I plan to go over 3.2, 3.9, 3.10, 3.11, 3.12, 3.14, parts of 3.15 and 3.18, and 3.21 in class, which should hopefully prepare you for the rest of the assignment. 

The remaining sections from Lay that we will cover this quarter are listed below. In-class coverage will be light, as in my judgement, there is little that is completely new, especially to students who have been simultaneously studying physics. There may be problems assigned from these sections of Lay for homework, but even if not, when we encounter these ideas in QM, I expect that you will be able to use Lay as a resource.

• Section 5.5: Complex Eigenvalues. In particular, “Eigenvalues and Eigenvectors of a Real Matrix That Acts on Cⁿ.” Note that Example 6 and the Practice Problem at the end of the section reinforce question 8 from LA/QM Exam 1.
• Section 6.1: Inner Product, Length, and Orthogonality. Aside from terminology, little in this section should be brand new to you. A possible exception is the discussion of Orthogonal Complements, which nicely adds to our previous discussions of important subspaces, and sets up for sections 6.2 and 6.3.
• Section 6.2: Orthogonal Sets. This section introduces some new ideas though I think little that is counter-intuitive.
• Section 6.3: Orthogonal Projections. Theorem 8 and the discussion surrounding Figure 3 are the most important parts of this section. Please skip Theorem 9 and its surrounding discussion.
• Section 6.4: The Gram-Schmidt Process. The theory developed in sections 6.2 and 6.3 culminates in this section, which provides an algorithm for producing an orthogonal basis for any (nonzero) subspace of Rⁿ. If you need to be strategic with your time, this is the section to concentrate on, referring to 6.2 and 6.3 as needed to support your understanding of the Gram-Schmidt Process. Please skip the section on “QR Factorization of Matrices”.
• Section 6.7: Inner Product Spaces. This section generalizes the results from earlier. Please skip the section on Best Approximation in Inner Product Spaces. The section “Two Inequalities” discusses the Cauchy-Schwarz Inequality, which is called the Schwarz Inequality in Townsend QM Ch. 3. The section “An Inner Product for C[a,b]” extends the ideas of inner products to the important case of C[a,b], continuous functions on the interval [a, b], as defined by eq. (5) in Example 7. This generalization of inner product from the discrete case to the continuous case will be very important for us when we cover Townsend QM Ch. 6 at the beginning of spring.

• You can find a sign-up sheet for our brief (10 minutes) Week 17 check-in meetings here. Meetings will be during the lunch break or after class on Tuesday and Thursday, and will be in Krishna’s office, Lab 2 room 3255.
• Note that you will need to be logged in to this site to access the sign-up sheet. If not logged-in, you can do so via this link.

At one of your clasSMate’s request, I’ve made Collaborative Revisions versions of the EM and LA/QM Exam 1 questions available. You can access them here; note that you have to be logged in to visit this page.

For Week 17, please read the following:

• EM: Griffiths 4.2, 4.3, 4.4.1
• LA: Lay 5.5
• QM: Townsend Ch. 3

• EM: Griffiths 4.2, 4.3
  
  • Reminder: video lectures here.
• LA: Lay 5.5
  
• QM: Townsend Ch.3.
  • Reminder: some information about possibly useful video lectures here.
• P/S: Microwave Interference; Physical (Fourier) Optics.

• Remember to use nb.mit.edu (info on how to access here) or check with faculty for alternatives. Complete first pass of reading and enter annotations (questions, responses, requests for class time) by Tue. 9am and update through week as need.

This week’s schedule. Except for Monday, this is a fairly regular week. Brief check-in meetings scheduled on Tuesday and Thursday during the lunch break and right after class, sign-up here; you’ll need to be logged-in to access the sign-up page (if not logged-in, you can do so via this link.)

• Mon. Feb. 20: Presidents’ Day holiday; no classes.
• Tue. Feb. 21:
  • 9:00 – 11:00: LA/QM *Exam Revisions due 9am*
  • 12:30 – 2:30: LA/QM
• Wed. Feb. 22:
  • 9:00 – 11:00: EM
  • 11:30 – 1:00: EM
• Thu. Feb. 23:
  • 9:00 – 11:00: P/S
  • 12:30 – 2:00: LA/QM
  • 2:00 – 3:00 EM & LA/QM Workshop

Joel will be available for tutoring in the Cave from 4 – 6 today.

In case you would like to get started on exam revisions early, here are the questions from:

• Linear Algebra/Quantum Mechanics Exam 1
• Electricity & Magnetism Exam 1

Guidelines:

• Revisions are due Week 7 Tuesday Feb. 21 at 9am. (Recall there are no classes on Mon. Feb. 20).
• You may choose to revise any exam questions you like, but must revise the entire question.
• Revisions to multiple choice/matching/ranking/fill in the blank type questions must include a full explanation (simply choosing the correct response is insufficient).
• Revisions should be of very high quality. Though not required to by typeset, they should be as carefully presented as you can make them be, given time constraints. Revised solutions must be perfectly clear, complete, and correct to be considered.
• You may use any resource to prepare your revised solution. HOWEVER…
  • …Revised solutions must display your personal understanding of the material. This means that you must personally understand every step that you write down, and individually produce the content of your final submitted revised solution.