Faculty of Arts & Sciences

This is the second term of a two-semester introductory sequence that uses a combination of analytic and numerical methods to understand physical systems, to analyze experimental data, and to compare data to models. Topics include electrostatics and magnetostatics, electromagnetic fields, optics [all topics illustrated with applications to current technological and societal challenges], and an introduction to the physics of many-body systems and their aggregate properties such as entropy, temperature and pressure. The course is aimed at second year students who have an interest in pursuing a concentration in the sciences and/or engineering. The course structure includes lecture, discussion and laboratory components.

Newtonian mechanics and special relativity. Topics include vectors; kinematics in three dimensions; Newton's laws; force, work, power; conservative forces, potential energy; momentum, collisions; rotational motion, angular momentum, torque; static equilibrium, oscillations, simple harmonic motions; gravitation, planetary motion; fluids; special relativity.

Electricity and magnetism. Topics include electrostatics, electric currents, magnetic field, electromagnetic induction, Maxwell's equations, electromagnetic radiation, and electric and magnetic fields in materials.

Forced oscillation and resonance; coupled oscillators and normal modes; Fourier series; Electromagnetic waves, radiation, longitudinal oscillations, sound; traveling waves; signals, wave packets and group velocity; two- and three-dimensional waves; polarization; geometrical and physical optics; interference and diffraction. Optional topics: Water waves, holography, x-ray crystallography, and solitons.

Newtonian mechanics and special relativity for students with good preparation in physics and mathematics at the level of the advanced placement curriculum. Topics include oscillators damped and driven and resonance (how to rock your car out of a snow bank or use a swing), an introduction to Lagrangian mechanics and optimization, symmetries and Noether's theorem, special relativity, collisions and scattering, rotational motion, angular momentum, torque, the moment of inertia tensor (dynamic balance), gravitation, planetary motion, and a quantitative introduction to some of the mind-bending ideas of modern cosmology like inflation and dark energy.

Primarily for selected concentrators in Physics, or in Chemistry and Physics, who have obtained honor grades in Physics 15 and a number of intermediate-level courses. The student must be accepted by some member of the faculty doing research in the student's field of interest. The form of the research depends on the student's interest and experience, the nature of the particular field of physics, and facilities and support available. Students wishing to write a senior thesis can do so by arranging for a sponsor and enrolling in this course.

Open to selected concentrators in Physics, Chemistry and Physics, and other fields who wish to do supervised reading and studying of special topics in physics. Ordinarily such topics do not include those covered in a regular course of the Department. Honor grades in Physics 15 and a number of intermediate-level courses are ordinarily required. The student must be accepted by a member of the faculty.

The goal of this tutorial is twofold. First, students will learn about a range of modern physics research topics from experts at Harvard as well as from one another. Every Wednesday evening a faculty member speaks on his/her area of research, preceded by assigned reading and a student presentation designed to introduce the basic physics, as well as important developments and burning problems at the frontiers of that particular research area. Second, the tutorial provides structured activities to help students develop practical skills for their future careers, expanding knowledge on unfamiliar subjects, participating in discussions, presenting and writing clearly about complex topics, and engaging in self and peer evaluation.

This course will introduce cosmology, the study of the large-scale evolution of the universe. Topics include the expanding universe; Friedmann-Robertson-Walker metrics; the evolution of the matter, radiation, and vacuum energy of the universe over time; evidence for dark matter; the Cosmic Microwave Background and its role in determining cosmological parameters; Big-Bang Nucleosynthesis; inflation and how it seeded the universe today; and the formation of structures like galaxies.

A lab-intensive introduction to electronic circuit design. Develops circuit intuition and debugging skills through daily hands-on lab exercises, each preceded by class discussion, with minimal use of mathematics and physics. Moves quickly from passive circuits, to discrete transistors, then concentrates on operational amplifiers, used to make a variety of circuits including integrators, oscillators, regulators, and filters. The digital half of the course treats analog-digital interfacing, emphasizes the use of microcontrollers and programmable logic devices (PLDs).

Applies elementary physics to the real world and fundamental phenomena, introducing estimation and calculational techniques that are commonly used by research physicists when addressing new problems. Emphasis is on developing physical intuition and the ability to do order-of-magnitude calculations. New physical concepts are introduced as necessary. Example topics: the Big Bang and searches for Earth-like exoplanets; material properties and phase transitions; masers, lasers, and the global positioning system; magnetic resonance imaging and physiology of major organs; Earth properties & human energy use. Example estimation techniques: dimensional analysis, commonly used concepts such as diffusion and the Bloch model, scaling laws, and symmetries and conservation laws.

Non-fossil energy sources and energy storage are important for our future. We cover four main subjects to which students with a background in physics and physical chemistry could make paradigm changing contributions: photovoltaic cells, nuclear power, batteries, and photosynthesis. Fundamentals of electrodynamics, statistical/thermal physics, and quantum mechanics are taught as needed to give students an understanding of the topics covered.

This course presents the underlying physics of modern medical diagnostic imaging techniques. We will explore the physics of diagnostic imaging from a unified electromagnetics' viewpoint ranging from a simple mapping of radiation attenuation coefficients in X-ray, to resonance absorption in a nuclear magnetic resonance (NMR) induced inhomogeneously broadened RF absorber. The bulk of the course will focus on the powerful technique of NMR imaging. Flexibility exists to vary the depth of each area depending on background and experience of the students.

We will discuss how theoretical and experimental tools derived from physics - e.g., statistical mechanics, fluid mechanics - have been used to gain insight into molecular and cellular biology including the structure and regulation of DNA, genomes, proteins, the cytoskeleton, and the cell. Students will gain an intensive introduction to biological systems, as well as physical and mathematical modeling.

Living organisms use sensory systems to inform themselves of the sights, sounds, and smells of their surrounding environments. Sensory systems are physical measuring devices, and are therefore subject to certain limits imposed by physics. Here we will consider the physics of sensory measurement and perception, and study ways that biological systems have solved their underlying physical problems. We will discuss specific cases in vision, olfaction, and hearing from a physicist's point of view.

In this class we introduce discuss physical and quantitative aspects of multi-scale organization in biology. We will study the mechanics, dynamics and statistical physics of embryonic development, and see how physics-based approaches are used in an attempt to understand cancer. We will look at collective animal behaviors, the dynamics of population, ecology and extinction. Finally, we will study models of evolution and population genetics.

Introduction to nonrelativistic quantum mechanics: uncertainty relations; Schrodinger equation; Dirac notation; matrix mechanics; one-dimensional problems including particle in box, tunneling, and harmonic oscillator; angular momentum, hydrogen atom, spin, Pauli principle; time-independent perturbation theory; scattering.

Introduction to path integrals, identical particles, WKB approximation, time-dependent perturbation theory, photons and atoms, scattering theory, and relativistic quantum mechanics.

This course will review the role of symmetries in quantum mechanics. Topics include atomic and molecular symmetries, crystallographic symmetries, spontaneous symmetry breaking and phase transitions, geometrical Berry phases, topological aspects of condensed matter systems. Mathematical basics of group theory will be taught as needed to give students an understanding of the topics covered.

Introduction to elementary particle physics. Emphasis is on concepts and phenomenology rather than on a detailed calculational development of theories. Starts with the discovery of the electron in 1897, ends with the theoretical motivation for the Higg's boson, and attempts to cover everything important in between. Taught partly in seminar mode, with each student presenting a classic paper of the field.

Fundamental ideas of classical mechanics including contact with modern work and applications. Topics include Lagrange's equations, the role of variational principles, symmetry and conservation laws, Hamilton's equations, Hamilton-Jacobi theory and phase space dynamics. Applications to celestial mechanics, quantum mechanics, the theory of small oscillations and classical fields, and nonlinear oscillations, including chaotic systems presented.

Aimed at advanced undergraduates. Emphasis on the properties and sources of the electromagnetic fields and on the wave aspects of the fields. Course starts with electrostatics and subsequently develops the Maxwell equations. Topics: electrostatics, dielectrics, magnetostatics, electrodynamics, radiation, wave propagation in various media, wave optics, diffraction and interference. A number of applications of electrodynamics and optics in modern physics are discussed.

Includes the use of coherent electromagnetic radiation to probe and control atomic systems, use of traps to isolate atoms, molecules, and elementary particles for studies of ultracold quantum degenerate matter and precision tests of the standard model; resonance methods. Goals of course include acquainting student with these and other modern research topics while providing the foundations of modern atomic, molecular and optical physics research.

Introduction to laser physics and modern optical physics aimed at advanced undergraduates. Review of electromagnetic theory and relevant aspects of quantum mechanics. Wave nature of light. Physics of basic optical elements. Propagation of focused beams, optical resonators, dielectric waveguides. Interaction of light with matter, introduction to quantum optics. Lasers. Physics of specific laser systems. Introduction to nonlinear optics. Modern applications.

Introduction to thermal physics and statistical mechanics: basic concepts of thermodynamics (energy, heat, work, temperature, and entropy), classical and quantum ensembles and their origins, and distribution functions. Applications include the specific heat of solids, black body radiation; classical and quantum gases; magnetism; phase transitions; propagation of heat and sound.

Students carry out three experimental projects selected from those available representing condensed matter, atomic, nuclear, and particle physics. Included are pulsed nuclear magnetic resonance (with MRI), microwave spectroscopy, optical pumping, Raman scattering, scattering of laser light, nitrogen vacancies in diamond, neutron activation of radioactive isotopes, Compton scattering, relativistic mass of the electron, recoil free gamma-ray resonance, lifetime of the muon, studies of superfluid helium, positron annihilation, superconductivity, the quantum Hall effect, properties of semiconductors. The facilities of the laboratory include several computer controlled experiments as well as computers for analysis.

This course gives a grounding in condensed matter physics, with an emphasis on solid state materials of practical utility. We give a physical & quantitative introduction to crystal structure, band structure, electrons, phonons, thermal properties, optical properties, and magnetic properties. We cover materials including metals, insulators, semiconductors, and superconductors. We include discussion of experimental techniques employed to measure material properties.

An introduction to general relativity: the principle of equivalence, Riemannian geometry, Einstein's field equation, the Schwarzschild solution, the Newtonian limit, experimental tests, black holes.

A survey of black holes focusing on the deep puzzles they present concerning the relations between general relativity, quantum mechanics and thermodynamics. Topics include: causal structure, event horizons, Penrose diagrams, the Kerr geometry, the laws of black hole thermodynamics, Hawking radiation, the Bekenstein-Hawking entropy/area law, the information puzzle, microstate counting and holography. Parallel issues for cosmological and deSitter event horizons will also be discussed.

Develops theoretical basis for modeling and quantitative analysis of biological problems. Emphasis on contemporary research topics, including molecular, cellular and tissue dynamics; development and differentiation; signal- and mechano-transduction; individuals, populations and environments.

Introduction to functional analytic methods relevant for problems in quantum and statistical physics. Properties of linear transformations on Hilbert space. Generators of continuous groups and semigroups. Properties of Green's functions and matrices. Uniqueness and non-uniqueness of ground states and equilibrium states. Heat kernel methods. Index theory, invariants, and related algebraic structure. The KMS condition and its consequences.

Semiclassical approaches to quantum systems provide both intuitive understanding of quantum processes and methods for calculations that are vastly simpler than full quantum mechanical simulations. Semiclassical methods are based on classical mechanics including interference and phases computed with classical actions. The course, based on a textbook being written by Prof. Heller (The Semiclassical Way to Quantum Mechanics) begins with a review of some salient features of classical physics, followed by an introduction to stationary phase integration and the Feynman Path Integral in the semiclassical limit, including time and energy domains, and the famous Trace Formula. This is followed by a number of widely useful techniques, such as generalized tunneling, applications to classically chaotic systems, semiclassical wave packet dynamics, WKB methods and uniformization. A number of "special topics" will then be taken up, including decoherence, certain forms of spectroscopy, and scattering theory of nanoscale devices.

An introduction to electronic circuit design intended to develop circuit intuition and debugging skills through daily design exercises, discussion and hands-on lab exercises. The approach is intensely practical, minimizing theory. Moves quickly from passive circuits to discrete transistors, then concentrates on operational amplifiers, used to make a variety of circuits including integrators, oscillators, regulators, and filters. The digital half of the course treats analog-digital interfacing, emphasizes the use of microcontrollers and programmable logic devices (PLDs).

Special relativity, relativistic field theories, gauge invariance, the Maxwell equations, conservation laws, time-independent phenomena, multipole expansions, electrodynamics and radiation theory, radiation from rapidly-moving accelerating charges, scattering and diffraction, and macroscopic averaged fields and propagation in matter. Additional topics may include relativistic particles with spin, coherent states, superconductors, accelerator physics, renormalization, and magnetic monopoles.

Three experimental projects are selected representing condensed matter, atomic, nuclear, and particle physics. Examples: experiments on pulsed nuclear magnetic resonance, microwave spectroscopy, optical tweezers, and non-linear optics, optical pumping, Raman scattering, scattering of laser light, nitrogen vacancies in diamond, neutron activation of radioactive isotopes, Compton scattering, relativistic mass of the electron, recoil free gamma-ray resonance, lifetime of the muon, studies of superfluid helium, positron annihilation, superconductivity, the quantum Hall effect, properties of semiconductors. The facilities of the laboratory include several computer controlled experiments as well as computers for analysis.

Recently, the Higgs particle was discovered experimentally by the ATLAS and the CMS Collaborations at CERN; it is the first spin-0 elementary particle ever observed. It is the purpose of this course to discuss various topics related to this particle.

Basic course in nonrelativistic quantum mechanics. Review of wave functions and the Schrodinger Equation; Hilbert space; the WKB approximation; central forces and angular momentum; electron spin; measurement theory; the density matrix; perturbation theory.

Potential topics include Heisenberg picture; time-dependent perturbations; inelastic scattering; electrons in a uniform magnetic field; quantized radiation field; absorption and emission of radiation; identical particles and second quantization; nuclear magnetic resonance; Feynman path integrals for quantum spins.

Introduction to relativistic quantum field theory. This course covers quantum electrodynamics. Topics include canonical quantization, Feynman diagrams, spinors, gauge invariance, path integrals, ultraviolet and infrared divergences, renormalization and applications to the quantum theory of the weak and gravitational forces.

A continuation of Physics 253a. Topics include: the renormalization group, implications of unitarity, Yang-Mills theories, spontaneous symmetry breaking, weak interactions, anomalies, and quantum chromodynamics. Additional advanced topics may be covered depending on time and interest.

Introduction to some of the tools for studying the exact nonperturbative dynamics of supersymmetric gauge theories, supergravity, and gauge/gravity duality.

The Standard Model of particle physics: theory and experimental implications. Topics include nonabelian gauge theory, spontaneous symmetry breaking, anomalies, the chiral Lagrangian, QCD and jets, collider physics and simulation, the Higgs at the LHC.

Basic principles of statistical physics and thermodynamics, with applications including: the equilibrium properties of classical and quantum gases, phase transitions and critical phenomena, as illustrated by the liquid-gas transition and simple magnetic models. Universality, scaling and renormalization group. Introduction to non-equilibrium physics.

Notions of topology have been invoked to clarify the properties of a variety of quantum systems and to classify the possible ground states of such systems. We shall explore in depth examples such as two-dimensional quantized Hall states, and topological insulators in two and three dimensions. Discussions will include effects of disorder and localization phenomena, and practical issues of measurement that may have only marginal relation to topological concepts.

Introduction to strongly interacting soft condensed matter and biophysical systems. We begin with the physics of cells and related single molecule experiments on bio-polymers such as DNA, RNA and proteins. A major part of the course will then focus on genetic engineering, and the non-equilibrium statistical dynamics of genetic circuits and neural networks.

Introduces the subject of quantum effects in electronic systems, including conductance fluctuations, localization, electron interference, and many-body effects such as the Kondo effect. This year, we will also focus on solid state implementations of quantum information processing systems.

Introduction to physics of quantum information, with emphasis on ideas and experiments ranging from quantum optics to condensed matter physics. Background and theoretical tools will be introduced. The format is a combination of lectures and class presentations.

Covers current advances in particle physics beyond the Standard Model. Topics could include supersymmetry, the physics of extra dimensions, experimental searches, including for T violation, and connections between particle physics and cosmology.

Explores an emerging interface involving strongly correlated systems in atomic and condensed matter physics. Topics include bosonic and fermionic Hubbard models, strongly interacting systems near Feshbach resonances, magnetism of ultracold atoms, quantum spin systems, low dimenstional systems, non-equilibrium coherent dynamics.

Introduction to modern atomic physics. The fundamental concepts and modern experimental techniques will be introduced. Topics will include two-state systems, magnetic resonance, interaction of radiation with atoms, transition probabilities, spontaneous and stimulated emission, dressed atoms, trapping, laser cooling of "two-level" atoms, structure of simple atoms, fundamental symmetries, two-photon excitation, light scattering and selected experiments. The first of a two-term subject sequence that provides the foundations for contemporary research.

Introduction to quantum optics and modern atomic physics. The basic concepts and theoretical tools will be introduced. Topics will include coherence phenomena, non-classical states of light and matter, atom cooling and trapping and atom optics. The second of a two-term subject sequence that provides the foundations for contemporary research.

Introduction to the perturbative formulation of string theories and dualities. Quantization of bosonic and superstrings, perturbative aspects of scattering amplitudes, supergravity, D-branes, T-duality and mirror symmetry. Also a brief overview of recent developments in string theory.

A selection of topics from current areas of research on string theory.

The course will give the reconstruction of relativistic quantum fields from Euclidean fields as well as the relation between representations of the Poincare group to those of Euclidean group. Related topics are reflection positivity and Osterwalder-Schrader quantization, and supersymmetry, some of which will be covered.

Electrical, optical, thermal, magnetic, and mechanical properties of solids will be treated based on an atomic scale picture and using the independent electron approximation. Metals, semiconductors, and insulators will be covered, with possible special topics such as superconductivity.

Theory of the electron liquid, Fermi liquid theory. Ferromagnetism of metals, BCS theory of superconductivity. Lattice models of correlated electrons: antiferromagnetism. Graphene: semi metals and quantum phase transitions. Non-fermi liquids in correlated metals.

Hands-on, experienced-based course for graduate students on teaching and communicating physics, conducted through practice, observation, feedback, and discussion. Departmental rules for teaching fellows, section and laboratory teaching, office hours, assignments, grading, and difficult classroom situations.