Weekly Agenda

Week 2: Basis, Metric, Maps

• Lecture 3: Basis vectors, change of basis

• Lecture 4: Differential operators as vectors (and a hint of differential geometry), special relativity example: what is the energy of an object as observed by someone in a different reference frame? 

Abstracting our notion of vectors and matrices. Matrices (tensors) as (multi-)linear maps acting on vectors. 

• Short homework (due Thursday, 4/7): Changing basis. Homework submission link.

• Long homework (due Thursday 4/14): Working with Vector Spaces. Matrix space, Gram–Schmidt procedure, rotations. Homework submission link. (same as last week)

Reading

• Laforest 2.5-2.6

• Shankar 1.2

• Stone and Goldbart, Mathematics for Physics, Ch. 10 [ebook via library]

Week 3: Gram-Schmidt, Intro to Eigensystems

• Lecture 5: Gram-Schmidt, by Ian Chaffey

• Lecture 6: Eigenvectors, Eigenvalues

Homework

Long Homework 1 is due this Thursday.

• Short Homework 3. Gram-Schmidt process. Due Thursday, Apr 13. Submission link.

• Long Homework 2: rotations/boosts, eigenvalues and eigenvectors, metrics and relativity. Due Thursday, April 28. Submission link.

• Explainer video. To be assigned shortly.

Week 4: Eigenstuff

• Lecture 7: Symmetric Matrices, characteristic equations

• Lecture 8: Determinants, traces. Finding eigenvalues and eigenvectors. Powers of matrices.

Homework

• Short Homework 4. Diagonal and symmetric matrices. Due Thursday, Apr 21. Submission link.

• Long Homework 2: rotations/boosts, eigenvalues and eigenvectors, metrics and relativity. Due Thursday, April 28. Submission link.

• Explainer video assignments. Submission link.

Week 5: Commutators, Complex Vector Spaces

• Lecture 9: Commutators, degenerate eigenvalues

• Lecture 10: "Spring theory." An application of eigenvalues to the normal modes of a system of springs. (See Lec 9 notes.)

Homework

Long Homework 2 is due this Thursday.

• Short Homework 5. Commutators, degenerate eigenvalues. Due Thursday 4/28. Submission link.

• Long Homework 3. Complex spaces, "infinite" dimensional spaces. Due Thursday, 5/12. Submission link.

• Explainer video assignments. Submission link. [to be assigned]

Week 6: Function Space, Complex Space

• Lecture 11: spring theory, discrete function space

• Lecture 12: derivatives on function space, locality of physical theories

Homework

• Short Homework 6. Commutators, degenerate eigenvalues. Due Thursday 5/5. Submission link.

Week 7: Function Space, Complex Space

• Lecture 13: Complex function space

• Lecture 14: Complex function space 

Homework

• Short Homework 7. Due Thursday, 5/12. Fourier space. Submission link.

• Long Homework 4. Due Thursday 5/26, adventures in function space. This one is a little hefty, but it is the culmination of a lot of what we've been working toward in this course. Submission link.

Week 8: Fourier Transforms

• Lecture 15: Fourier transforms; see also Addendum for Problem 2 of the long homework

• Lecture 16: Fourier transforms, a bird's eye view

Homework

• Short Homework 8. Due Thursday, 5/19. Fourier space. Submission link.

• Long Homework 4. Due Thursday 5/26, adventures in function space. This one is a little hefty, but it is the culmination of a lot of what we've been working toward in this course. Submission link.

  • Addendum for Problem 2 (some hints on the normalization)

Week 9: Aspects of Quantum Mechanics

• Lecture 17: intro to two state quantum systems, Stern-Gerlach, BB84 quantum key distribution protocol.

  • The discussion of the Stern-Gerlach experiment and the BB84 protocol comes from Quantum Computing for Everyone by Chris Bernhardt. 

• Lecture 18: continuing BB84 from Lec 17. Lecture 18 notes include additional topics on entanglement, representation of groups, and more examples for Fourier analysis. 

  • 3Blue1Brown video on Fourier transforms. 

  • Mathematica example for Fourier series

  • Howard Georgi's textbook on the physics of waves, an excellent reference for the micro- and macrophysics of waves (including Fourier series). The book is surprisingly deep for something that spends so much time on coupled harmonic oscillators; it may be fun summer reading.

  • An update on quantum cryptography from DefCon 2019

Homework

• Short HW 9: Stern-Gerlach basics. Submission link.

• Long Homework 4. Due Thursday 5/26, adventures in function space. This one is a little hefty, but it is the culmination of a lot of what we've been working toward in this course. Submission link.

  • Addendum for Problem 2 (some hints on the normalization)

Week 10: entanglement, tensor products

• Lecture 19: Tensor products and entanglement, the big idea of Shor's algorithm and quantum computing. Notes: see Lec 18 notes, additional notes.

  • You may enjoy reading Shor's original paper on factoring with a quantum computer. The language is different from what we're used to in physics, but you should recognize everything in section 2.

  • An eminently readable book on computation for physicists is The Nature of Computation by Moore and Mertens. You should be able to jump into the chapter on quantum computation near the back of the book. 

  • Those who are interested in quantum computation should consider applying for the Los Alamos National Laboratory Quantum Computation School next summer.

• Lecture 20: Machine learning as linear algebra, and why it's not really linear algebra.

  • Deep learning has become one of the really big ideas in science and engineering over the last decade. An excellent reference on the theory of deep learning (rather than the implementation) is a new textbook written by physicists, The Principles of Deep Learning Theory.

  • 3Blue1Brown has an excellent introduction to how neural networks work.

  • Those who are interested in machine learning in science may want to consider the Los Alamos Applied Machine Learning Summer Research Fellowship next summer.

Homework

• Short HW 10: Stern-Gerlach basics. Submission link.