Final Program Description and Course Equivalencies

Faculty: Krishna Chowdary, Ph.D.,  Brian L. Walter, Ph.D., John Caraher, Ph.D.

Physical Systems & Applied Mathematics was a full year interdisciplinary study of junior-senior level physics (classical mechanics, electrodynamics, quantum mechanics, and advanced lab) closely integrated with related sophomore-level applied mathematics (differential equations, linear algebra, and multivariable and vector calculus, along with an integrated math lab in the computer algebra system Mathematica). Students opted to study some or all of the available program subjects. Evaluation of student achievement was based on: quizzes, exams, and revisions; problem sets and contributions to collaborative solutions sets; writing assignments; and class participation.

In addition to content coverage described below, program learning goals included: Creating an intentionally inclusive and anti-bias learning environment; Improving ability to articulate and assume responsibility for personal work; Strengthening collaborative skills and the ability to respond in useful ways to the work of colleagues; Improving skills in clear communication of mathematical and scientific ideas, both orally and in writing; Improving reading of technical texts to develop both conceptual understanding and procedural skills; Developing increasingly sophisticated mathematical models to describe and explain physical systems; Using multiple representations to gain a firm understanding of the concepts and procedures of differential equations, linear algebra, and multivariable and vector calculus; Developing deep conceptual understanding and sophisticated problem-solving abilities related to classical mechanics, electricity & magnetism, and quantum mechanics; Using Mathematica to visualize and solve problems to gain insight into the related mathematics; Developing insight into the fundamental interplay between the experimental, computational, and theoretical aspects of physics through exposure to a variety of advanced laboratory experiments and their historical and social contexts and implications.

Classical Mechanics: Chapters 1-9, 11, and 13 of Classical Mechanics (Taylor) were covered in winter and spring, using differential equations and multivariable and vector calculus covered in fall. Topics included: Newtonian mechanics in different coordinate systems; projectile and charged particle motion; momentum and angular momentum; energy; oscillations (including Fourier series); calculus of variations; Lagrangian mechanics (including the method of multipliers and constraints); central-force problems; non-inertial reference frames; coupled oscillations and normal modes; and Hamiltonian mechanics. Students completed 17 homework assignments totaling 120 problems and contributed to collaborative solutions sets for each homework assignment; in spring, the 8 solution set contributions were type-set. Students took 9 hour-long exams (one of which was take-home), completed a substantial cumulative final exam, and could choose to submit exam revisions.

Electrodynamics: Chapters 1-5, 7-9, and 12 of Introduction to Electrodynamics, 4th Edition (Griffiths) were covered in winter and spring, building heavily on foundational work in multivariable and vector calculus from fall. Topics included: vector analysis; electrostatics in vacuum and matter and magnetostatics in vacuum (including forces, fields, and potentials); special techniques (method of images, Laplace’s equation, separation of variables); Maxwell’s equations; energy and momentum in fields; electromagnetic waves; and special relativity. Students completed 17 homework assignments totaling 125 textbook problems, and in winter made contributions to 9 collaborative solutions sets. Students submitted two lab reports, one on determining the magnetic moment of a permanent magnet (based on TeachSpin, Inc.’s Magnetic Force Apparatus) and the other on Faraday’s law. Students took 9 hour-long exams (two of which were take-home), completed a substantial cumulative final exam, and optionally submitted exam revisions.

Quantum Mechanics: Chapters 1-7 and 9-11 of A Modern Approach to Quantum Mechanics, 2nd Edition (Townsend) were covered in winter and spring, relying heavily on foundational work in linear algebra from fall. Using a spins-first approach, topics included: Stern-Gerlach experiments; Dirac notation; rotation of basis states and matrix mechanics; operators and expectation values; angular momentum; the Schrödinger equation and time evolution; systems of spin-1/2 particles; wave mechanics in one dimension for barriers, finite and infinite square wells, steps, and Gaussian wave packets; the one-dimensional harmonic oscillator with raising and lowering operators and in the position basis; orbital angular momentum and the rigid rotator; bound states for central potentials including the hydrogen atom and three-dimensional wells and harmonic oscillators; and time-independent perturbation theory. Students completed 15 homework assignments totaling 120 problems, and in winter made contributions to 8 collaborative solutions sets. Students completed 13 take-home quizzes that were reviewed during in-class workshops; revised quizzes were then submitted for faculty review. Students took a substantial cumulative final exam, and had the option to submit an exam revision.

Seminar in Experimental Physics: In winter, a seminar on experimental physics surveyed a variety of advanced physics lab experiments and apparatus that covered some of the breadth of physics. Students read selections from apparatus manuals, primary source literature, historical accounts, textbook excerpts, and other technical material. Topics/apparatus included: Brownian motion; Franck-Hertz experiment; gyroscopes; laser cavities; magnetic domains; mechanical chaos; microwave interference;  physical (Fourier) optics; quantum optics (single photon interference); speed of light; superconductivity and magnetic susceptibility; and TeachSpin’s Faraday Rotation, Fourier Methods, Pulsed NMR, and Quantum Analogs. Students produced short Instrument Summaries for each of the 15 instruments/experiments encountered, which they discussed during five seminars, and revised for faculty review. Students also read Chapters 1-5 and 7-12 in Taylor’s An Introduction to Error Analysis (2nd Edition), and completed and self-corrected 83 short exercises. Students participated in two measurement focused lab sessions, intended to provide an overview/review of basic electronic components, multimeters, function generators, and oscilloscopes.

Advanced Lab Projects: In spring, students chose to focus on one or two pieces of apparatus introduced in the winter seminar in experimental physics in order to experience the fundamental interplay between the experiment, computation, and theory. Students developed personalized learning goals to which they were held accountable and were required to include some technical writing. Students gave weekly presentations, demonstrated their chosen instruments as part of the 16th annual Evergreen Science Carnival, presented a final conference-style talk to the entire class, and submitted their final technical writing for faculty review.

Differential Equations: The text used was Boas’s Mathematical Methods in the Physical Sciences, 3rd edition, from which Chapters 7, 8, and parts of 12 and 13 were covered in fall. Topics included ordinary differential equations (separable equations, linear first-order equations, second-order linear equations with constant coefficients, and other techniques, including the Laplace transform), Fourier series and transforms (including equations of simple harmonic motion, Fourier coefficients, and the sine-cosine and complex form of Fourier series), series solutions of differential equations (including Legendre’s equation and Legendre polynomials), and partial differential equations (including Laplace’s equation, the heat flow equation, and the wave equation). Students submitted 9 homework assignments and contributed to weekly collaborative solutions sets. They also took 4 in-class quizzes and one in-class cumulative final exam, and optionally submitted quiz and exam revisions.

Linear Algebra: The text used was Strang’s Introduction to Linear Algebra, 5th edition, from which chapters 1-6 and 8 were covered in fall. Topics included: vectors, solving systems of linear equations, matrix operations, vector spaces, orthogonality, determinants, eigenvalues & eigenvectors, and linear transformations. Students submitted 9 homework assignments and contributed to weekly collaborative solutions sets. They also took 4 in-class quizzes and one in-class cumulative final exam, and optionally submitted quiz and exam revisions.

Multivariable and Vector Calculus: Texts used in fall were Boas’s Mathematical Methods for the Physical Sciences, 3rd ed. (chapters 4-6) and Schey’s Div, Grad, Curl, and All That, 4th ed. Topics in multivariable calculus included partial derivatives, multiple integrals and their applications (finding extrema and gradients, mass, density, center of mass, and moments of inertia) in rectangular, cylindrical, and spherical coordinates. Topics in vector calculus included gradient, divergence, and curl, and line integrals and surface integrals for scalar-valued functions and vector fields, culminating in the fundamental theorems of vector calculus (the Gradient Theorem for Line Integrals, Green’s Theorem, the Divergence Theorem, and the Curl (Stokes’) Theorem). Students submitted 9 homework assignments totaling 121 problems and contributed to weekly collaborative solutions sets. They also took 4 in-class quizzes and one in-class cumulative final exam, and optionally submitted quiz and exam revisions.

Integrated Math Lab: In the fall quarter, students completed a series of investigations primarily using the computer algebra system Mathematica that supported and extended their studies of differential equations and multivariable and vector calculus. Students learned to use the program to visualize and solve problems, while also exploring and learning math concepts. Students completed 17 lab exercises to learn how to: visualize functions in two or three dimensions, including contour plots and vector fields; compute derivatives and integrals of multivariable functions; solve differential equations analytically and numerically; produce simple code; and apply these tools widely to differential equations and multivariable and vector calculus content.

Seminar: In fall, students engaged with a series of readings and associated writing exercises designed to help them reflect on the structure and purposes of higher education in general and physics and math education in particular, to help them develop their plans for their own academic paths, and to update drafts of their annual academic statements.

Course Equivalencies (*upper-division science credits)

  • *8 – Classical Mechanics I and II
  • *8 – Electrodynamics I and II
  • *8 – Quantum Mechanics I and II
  • *4 – Seminar in Experimental Physics
  • *4 – Advanced Lab Projects
  • *4 – Differential Equations
  • *4 – Linear Algebra
  • *4 – Multivariable and Vector Calculus

Final Conferences

  • I’m looking forward to our Final Conferences this week.
  • I have your graded CM & EM Final Exams along with your CM, EM, and QM Exam Revisions ready for you to pick up at your conference. You can also pick up your final set of PSNs at your conference as well.
  • Being sick slowed me down a great deal. I’ll be prepared at your conferences to discuss earned credit, but won’t have made much progress in drafts of evaluations. We’ll still be able to have good conversations about your work this year.

End-of-Program Reminders

  • CM, QM Final Exam Revisions are due by 5pm Fri. Jun. 7 to Krishna’s office Lab 2 3255 (originally due time was 3pm).
  • EM Final Exam Revisions are due by 9am Mon. Jun. 10 to Lab 2 3255.
  • Some notes about Exam Revisions are here.
  • ALP Final Papers are due by 5pm Mon. Jun. 10 via email to Krishna and John.
  • All other materials (quiz, quixam, PSN + Log, etc. re-submissions or submissions) should already have been turned in.
  • You can sign up for Final Conferences here.
  • Self-Evaluations are due to be shared via your by noon the day before your Conference.

Submissions or Re-submissions

  • Ungraded quizzes, quixams, and revisions were returned to you for study purposes. Please turn back in any ungraded quizzes, quixams, or revisions to the appropriate folders in the Cave: orange folders for CM, blue folders for EM, and purple folders for QM.
  • If a quiz, quixam, or revision was fully graded, please do not re-submit. For the most part, this is unambiguous, but there may be one or two that I started grading and didn’t complete grading when returned to you for studying.
  • If you are submitting a QM Quiz or a CM or EM Revision for the first time, please mark it with “Late” when you place it in the appropriate folder.
  • Please alphabetize when you turn in to each folder.
  • Please also submit CM, EM, and QM PSNs + Logs for final checks.
  • All of these need to be submitted by 9:30am Thursday June 6.

Sign-up for Final Conferences

  • Final Conferences are scheduled for Wed. Jun. 12 and Thu. Jun. 13.
  • You can sign up for a Final Conference here.
  • Conference lengths for graduating seniors are 40 minutes so we can discuss Final Academic Statements. Conference lengts for continuing students are 20 minutes.
  • Please submit your final program Self-Evaluation by noon the day prior to your Conference via your Make sure you have shared it with faculty so that I can access it.

Advanced Lab Projects Final Presentations, Wed. June 5, 9am – 1pm

  • Students who worked on Advanced Lab Projects in spring quarter will be presenting on their work from 9am – 1pm on Wednesday June 5, in the Cave, as previously announced.
  • All students in PSAM are expected to attend as their schedule allows. Students who have other commitments should have been proactive in contacting me.
  • Your classmates have worked hard, done some cool things, and have learned a great deal that they are eager to share with you!

EM Final Exam!!!

Here is the Electricity & Magnetism Final Exam!

Notes/Clarifications/Corrections: (updated 4:15pm Tue. Jun. 4)

  • Remembers to limit your time on the exam to 6 hours total. Those need not be 6 consecutive hours. Exams are due directly to me in the Cave at 9am Wed. June 5.
  • I apologize for the broken link to the exam; thanks for those who alerted me. The link was fixed by 5pm Mon. Jun. 3.
  • I don’t think anyone will need to adjust your work if you’ve already completed Problem 13, but we should think of part b) as follows:
    • 13b1) Set up the definite integral(s) that would allow you to find the total angular momentum vector of the fields (with respect to the z axis of the shells).
    • 13b2) Evaluate the definite integral(s). You may use technology to help you with the evaluation(s) (though you can do them all by hand).
  • Problem 4g: outside the shell should say r > b, not r > a.
  • Problem 8c: With the given charge distribution, you wind up with one integral that is best evaluated with technology (the other integral(s) can be evaluated without technology). Get as far as you can on the ones you can evaluate.

EM Exam info

  • A reminder that you can use the inside covers of Griffiths and Taylor for the CM Final Exam. Those are available in the Cave for you to pick up, if you haven’t picked them up already. You should keep them for use on the CM and EM Exams.
  • In addition to the inside covers of Griffiths and Taylor, you are also able to use a personally prepared 8.5 inch by 11 inch note sheet (both sides) that you have prepared prior to starting the CM Exam.
  • The EM Exam will be available on-line starting at 3pm on Monday June 3 for those of you taking the at-home option. If you are planning to take the exam in person in the Cave on Tuesday June 4, please try to let me know in advance so I can have copies on hand. If you decide to take the exam in person and haven’t emailed in advance, that is just fine – there will be a very slight delay as I print an exam copy for you. A reminder that I will provide lunch for those of you who take the exam in person.
  • EM Exams are due directly to me in the Cave at 9am Wednesday June 5.

EM Collected Quixams and Week 30 Seminar Reading

  • I’ve collected together all the EM Quixams into a single document (with updates/corrections to questions that were originally flawed); this is available as a paper copy in the front of the Cave on the side table.
  • Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” is also available for pick up in the front of the Cave on the side table. In a previous discussion, we expressed interest in discussing this paper during our lunch time conversation at our final program potluck/celebration on Thursday June 6. It seems a great way to wrap up our time together.