Faculty: Krishna Chowdary and Vauhn Foster-Grahler

*Physical Systems & Applied Mathematics *was a full year interdisciplinary study of upper-division physics (classical mechanics, electrodynamics, quantum mechanics, and advanced lab) closely integrated with related applied mathematics (differential equations, linear algebra, and multivariable and vector calculus). Students could choose partial credit options or to switch out some element of the program for independent study in computer programming. Evaluation of student achievement was based on: quizzes, exams, and revisions; homework assignments; written and oral presentations; writing assignments; and workshop participation.

In addition to content coverage, program learning goals included: 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 conceptual understanding and procedural skills; Developing and utilizing increasingly sophisticated mathematical models that 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 & vector calculus; Developing deep conceptual understanding and sophisticated problem-solving abilities related to classical mechanics, electricity & magnetism, and quantum mechanics; Using a computer based algebra system to visualize and solve complex problems in math and physics; Developing insight into the fundamental interplay between the experimental, computational, and theoretical aspects of physics through exposure to a variety of advanced laboratory experiments.

**Electrodynamics**: All 12 chapters of *Introduction to Electrodynamics, 4th Edition* (Griffiths) were covered in winter and spring, building heavily on foundational work in multivariable and vector calculus in fall. Topics included: vector analysis; electrostatics and magnetostatics in vacuum and matter (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; time-delayed potentials and fields; radiation; and special relativity. Students completed 18 homework assignments totaling 180 textbook problems; for each assignment, students submitted one of their problem towards a collaborative solution set. In winter, students took 8 quizzes, a midterm exam, and a final exam. In spring, students took 5 exams.

**Classical Mechanics**: Chapters 1 – 9, 11, and 13 of *Classical Mechanics* (Taylor) were covered in fall, closely integrated with differential equations and multivariable and vector calculus. Students learned to use *Mathematica* for a wide variety of purposes, including visualization and to solve differential equations and integrals (analytically and numerically). 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 16 homework assignments totaling 94 problems. In addition, students produced high-quality solutions (either typeset or presented as narrated screencasts or videos) for 4 “three-star” level problems (Taylor’s challenge level problems) which were submitted for peer review and then revised for faculty. Students took 4 quizzes, a midterm exam, and a final exam.

**Quantum Mechanics**: Chapters 1 – 7 and 9 – 11 of *A Modern Approach to Quantum Mechanics, 2nd Edition* (Townsend) were covered in winter and spring; in winter, this was integrated closely with linear algebra. 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, the EPR paradox and Bell’s inequalities; 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 132 problems; for each assignment, students submitted one of their problem towards a collaborative solution set. In winter, students took 8 quizzes, a midterm exam, and a final exam. In spring, students took 5 exams.

**Advanced Lab Projects**: 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 apparatus manuals and found relevant primary source literature, historical accounts, textbook excerpts, and other technical material, which they read, annotated, and prepared for a weekly seminar conversation. Topics/apparatus included: Franck-Hertz experiment; gyroscopes; magnetic domains; mechanical chaos; microwave interference; physical (Fourier) optics; speed of light; superconductivity and magnetic susceptibility; and TeachSpin’s Faraday Rotation, Fourier Methods, Pulsed NMR, and Quantum Analogs. In spring, students chose to focus on one or two pieces of apparatus introduced in winter in order to experience the fundamental interplay between the experiment, computation, and theory. Students developed personalized learning goals which they were held accountable to and were required to include some technical writing. Students gave weekly presentations, culminating in a final presentation to the entire class and submitted their final technical writing for faculty review.

**Differential Equations, Multivariable and Vector Calculus**: Multivariable and Vector Calculus (*Calculus: Multivariable, 6th Edition *(Hughes-Hallett, et.al.), supplemented by *Div, Grad, Curl, and All That, 4th ed* (Schey)) was a rigorous study of the calculus of functions of several variables. Topics included Gradient Fields, Stokes’ Theorem, Green’s Theorem, Divergence, Curl, Line Integrals, calculus with Polar, Cylindrical and Spherical Coordinates, Directional Derivatives, operations with the del operator, along with the support skills of partial differentiation and multiple integration.

Differential Equations (*Elementary Differential Equations and Boundary Value Problems, 10th Edition *(Boyce and DiPrima)) was a rigorous study of initial value problems and boundary value problems. Topics included: First, second, and higher order linear differential equations, boundary value problems (heat conduction in a rod, wave equation), and Fourier Series. Most of the time was spent looking at homogeneous equations with constant coefficients. Students were asked to solve problems in a variety of ways and to identify patterns in solutions including complex and repeated roots to second order, linear, homogeneous equations. Students found and interpreted fundamental sets of solutions and used the Wronskian to determine linear independence.

Students investigated problems symbolically, graphically, and, verbally. Mathematica was used to support visualization of functions in space, direction fields, and integral curves.

**Linear Algebra**: Chapters 1 – 6 of Linear Algebra and its Applications, 5th Edition (Lay, et.al.) were covered in winter, integrated very closely with quantum mechanics. Emphasis was on concepts, procedures, and applications of systems and solutions for linear equations, and topics included: linear transformations; matrix algebra; the Invertible Matrix Theorem; determinants; vector spaces, including null space and column space; change of basis; eigenvalues and eigenvectors; and orthogonality. Students completed 6 homework assignments totaling 100 problems; for each assignment, students submitted several of their problems towards a collaborative solution set. Students took 8 quizzes, a midterm exam, and a final exam.

Suggested Course Equivalencies (*upper-division science credits)

- *12 – Electrodynamics
- *8 – Classical Mechanics
- *8 – Quantum Mechanics
- *8 – Advanced Lab Projects
- *4 – Differential Equations
- *4 – Linear Algebra
- *4 – Multivariable and Vector Calculus