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