# Physics

Director of undergraduate studies: Simon Mochrie, 68C SPL, 436-4809, dus.physics@yale.edu; physics.yale.edu/academics/undergraduate-studies

#### FACULTY OF THE DEPARTMENT OF PHYSICS

**Professors **†Charles Ahn, Yoram Alhassid, Thomas Appelquist, †Charles Bailyn, O. Keith Baker, Charles Baltay, Sean Barrett, Cornelius Beausang (*Adjunct*), †Hui Cao, Richard Casten (*Emeritus*), †Richard Chang (*Emeritus*), †Paolo Coppi, David DeMille, †Michel Devoret, Bonnie Fleming, †Marla Geha, Steven Girvin, Leonid Glazman, John Harris, Karsten Heeger, †Victor Henrich, Jay Hirshfield (*Adjunct*), †Jonathon Howard, Francesco Iachello, †Sohrab Ismail-Beigi, Steven Lamoreaux, Simon Mochrie, Vincent Moncrief, †Priyamvada Natarajan, Peter Parker (*Emeritus*), †Daniel Prober, Nicholas Read, Jack Sandweiss (*Emeritus*), †Robert Schoelkopf, Ramamurti Shankar, Witold Skiba, †A. Douglas Stone, Paul Tipton (*Chair*), Thomas Ullrich (*Adjunct*), C. Megan Urry, †Pieter van Dokkum, †John Wettlaufer, Michael Zeller (*Emeritus*)

**Associate Professors **Helen Caines, Sarah Demers, †Thierry Emonet, Walter Goldberger, Jack Harris, Daisuke Nagai, †Corey O'Hern, Nikhil Padmanabhan, †Hongxing Tang

**Assistant Professors **†Murat Acar, †Eric Michael Brown, †Damon Clark, †Liang Jiang, Reina Maruyama, David Moore, David Poland, †Peter Rakich

**Senior Lecturer **Sidney Cahn

**Lecturers **Stephen Irons, Rona Ramos, Adriane Steinacker

†A joint appointment with primary affiliation in another department.

Physics forms a foundation for all other sciences. The various undergraduate courses and degree programs offered by the Physics department provide students with a thorough preparation in physics for any career, as well as the general background in physics that should be part of a liberal education. The department offers four different introductory sequences and two degree programs. Also offered are introductory courses that fulfill the science and quantitative reasoning distributional requirements and are appropriate for non–science majors. Combined majors are available in Mathematics and Physics, Astronomy, Physics and Philosophy, and Physics and Geosciences. Applied Physics is a closely related major.

**Introductory courses with no calculus requirement** Physics courses numbered 120 or below are for students with little or no previous experience in physics who do not plan to major in the natural sciences. These courses have no college-level mathematics requirement and do not satisfy the medical school requirement.

**Introductory calculus-based lecture sequences**

- PHYS 170, 171 is for students with little background in physics and mathematics who will probably not major in the physical sciences but who may be interested in medical and biological sciences. Knowledge of differential and integral calculus at the level of MATH 112 or equivalent is a prerequisite. MATH 115 should be taken concurrently with PHYS 171.
- PHYS 180, 181 is for students with some previous background in physics and mathematics who plan to major in the physical sciences. Calculus at the level of MATH 112 is a prerequisite; MATH 115 and 120 should be taken concurrently.
- PHYS 200, 201 is for students with a strong background in mathematics and physics who plan to major in the physical sciences. Calculus at the level of MATH 115 is presumed. MATH 120 and either MATH 222 or 225 are typically taken concurrently.
- PHYS 260, 261 is intended for students who have had excellent training in and have a flair for mathematical methods and quantitative analysis; a solid foundation in physics is required. One of MATH 120, ENAS 151, PHYS 301, or MATH 230, 231 or equivalent should be taken concurrently with PHYS 260, 261.

If students have the appropriate mathematics background, they are advised to take a calculus-based physics course. Sir Isaac Newton developed calculus while trying to describe the world around him; it is the natural language of physics. Students enrolled in one of the calculus-based introductory courses will be invited to a series of Chairman's Teas, which provide an opportunity to discuss topics on the frontiers of physics with faculty and peers. Completion of a calculus-based course also prepares students for the 340-level series of advanced physics electives, which cover special topics of interest to both majors and nonmajors.

A guide to selecting physics courses is available to aid in course selection. Questions about placement should be addressed to the director of undergraduate studies.

**Introductory laboratories **Two different introductory laboratory sequences are offered: PHYS 165L, 166L, and PHYS 205L, 206L. Each of these laboratory courses earns one-half course credit. Students normally take the laboratory courses associated with the introductory physics sequence in which they are enrolled. Students should register for a section of the appropriate laboratory course during the first week of classes by logging onto Classes*v2.

- PHYS 165L, 166L is an introductory laboratory for those students interested in the biological sciences and medicine, but without a strong high school physics laboratory preparation. Related lecture courses are PHYS 170, 171, and PHYS 180, 181.
- PHYS 205L, 206L is for students who plan to major in the physical sciences. Related lecture courses are PHYS 180, 181, PHYS 200, 201, and PHYS 260, 261. Students who take the lecture courses in freshman year are advised to start this laboratory sequence with PHYS 205L in the spring of freshman year or in the fall of sophomore year.

**Advanced electives **A series of 340-level electives explores special topics of interest to both majors and nonmajors. The electives are open to any student in Yale College who has completed a year of introductory calculus-based physics (PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261). The offerings for 2016–2017 include PHYS 343, Gravity, Astrophysics, and Cosmology and PHYS 344, Quantum and Nanoscale Physics.

**Major degree programs **Two different majors are offered in Physics: the B.S. and the B.S. with an intensive major. Students in either program acquire advanced training in physics, mathematics, and related topics through the core courses. They use electives to design individualized programs with more depth or breadth, depending on their needs and interests. Both degree programs require research experience through PHYS 471 and 472—one term for the B.S. degree and two terms for the B.S. degree with an intensive major. Both programs are excellent preparation for a wide variety of postgraduate activities, including professional school in business, law, or medicine; graduate school in engineering or other sciences; or careers in business, consulting, financial services, government service, or teaching.

The B.S. program with an intensive major is distinguished by depth of study in advanced physics courses and prepares students to study physics or closely related physical sciences in graduate school. The director of undergraduate studies can help students in the B.S. program prepare for graduate school in physics by recommending appropriate electives to supplement the core courses.

**Credit/D/Fail courses **Courses taken Credit/D/Fail may not be counted toward the requirements of either major.

**B.S. degree program** The prerequisites are an introductory lecture course sequence with a mathematics sequence equivalent to, or more advanced than, the corequisite of the physics sequence. The following options are appropriate: PHYS 170, 171 with MATH 112, 115; or PHYS 180, 181 with MATH 115, 120; or PHYS 200, 201 with MATH 120 and either 225 or 222; or PHYS 260, 261 with MATH 120, ENAS 151, PHYS 301, or MATH 230, 231 or equivalent. In addition, the laboratory sequence PHYS 205L, 206L or PHYS 165L, 166L is required. Students who take these physics and mathematics courses starting in their freshman year may satisfy the prerequisites by the middle of their sophomore year. Students who begin taking physics courses in their sophomore year may also complete either the standard or the intensive major. Students are advised to take mathematics courses throughout their freshman year at the appropriate level.

Eight courses are required beyond the prerequisites, including the senior project. Students must take a mathematics course at the level of, or more advanced than, PHYS 301. Three courses at the core of the major involve advanced study of fundamental topics common to all branches of physics, and must be taken in order. The first two, PHYS 401 and 402, pertain to advanced classical physics (mechanics, statistical physics and thermodynamics, and electromagnetism), and the third, APHY 439 or PHYS 440, covers quantum mechanics. Three advanced elective courses are also required. Suitable advanced courses include the PHYS 340-level electives, an advanced laboratory such as PHYS 382L, and 400-level courses in Physics. Students may also find suitable advanced courses in other departments in the sciences, engineering, and mathematics. Courses taken to satisfy these requirements must be approved by the director of undergraduate studies. In order to pursue their individual interests in sufficient depth, many students choose to take more than the required number of advanced courses.

**Senior requirement for the B.S. degree program **The senior requirement for the standard B.S. degree is fulfilled by receiving a passing grade on a one-term research project in PHYS 471 or 472 or equivalent. Students should consult the director of undergraduate studies for further information.

**B.S. degree program, intensive major **The prerequisites for the B.S. degree with an intensive major are the same as for the standard program. Ten courses are required beyond the prerequisites, including the senior project. Students must take a mathematics course at the level of, or more advanced than, PHYS 301. Five courses at the core of the major involve advanced study of fundamental topics common to all branches of physics. Three of the courses pertain to advanced classical physics: mechanics (PHYS 410), statistical physics and thermodynamics (PHYS 420), and electromagnetism (PHYS 430). Two other courses incorporate quantum mechanics (PHYS 440 and 441). These courses must be taken in order because the ideas build progressively: PHYS 410 precedes 440, which precedes 441, 420, and 430.

Because experiment is at the heart of the discipline, the intensive major requires one term of advanced laboratory (PHYS 382L or equivalent) and at least two terms of independent research (PHYS 471, 472 or equivalent). One advanced elective course is required to complete the program. Suitable advanced courses include the PHYS 340-level electives and 400-level courses in Physics. Students may also find suitable advanced courses in other departments in the sciences, engineering, and mathematics. Courses taken to satisfy these requirements must be approved by the director of undergraduate studies. In order to pursue their individual interests in sufficient depth, many students choose to take more than ten advanced courses.

**Senior requirement for the B.S. degree program, intensive major **The senior requirement for the intensive major is fulfilled by receiving a passing grade on a two-term research project in PHYS 471, 472 or equivalent. Students should consult the director of undergraduate studies for further information.

**Sequence of courses **For both the standard B.S. degree and the B.S. degree with an intensive major, students are advised to begin the program in their freshman year to allow the greatest amount of flexibility in course selection. It is possible, however, to complete either program in a total of six terms, as illustrated below.

A program for a student completing the Physics B.S. in three years might be:

Freshman or Sophomore | Sophomore or Junior | Senior |
---|---|---|

PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261 | PHYS 206L | APHY 439 or PHYS 440 |

PHYS 205L | PHYS 301 | PHYS 471 or 472 |

Mathematics corequisites | PHYS 401 | Two advanced electives |

PHYS 402 | ||

One advanced elective |

A program for a student completing the intensive major in three years might be:

Freshman or Sophomore | Sophomore or Junior | Senior |
---|---|---|

PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261 | PHYS 206L | PHYS 441 |

PHYS 205L | PHYS 301 | PHYS 420 |

Mathematics corequisites | PHYS 410 | PHYS 430 |

PHYS 440 | PHYS 471 | |

PHYS 382L | PHYS 472 | |

One advanced elective |

**Approval of programs **All Physics majors in the sophomore, junior, and senior classes must have their programs approved by the director of undergraduate studies. Freshmen and undeclared sophomores who are interested in Physics or related majors are encouraged to meet with the director of undergraduate studies to discuss their questions and proposed programs.

#### REQUIREMENTS OF THE MAJOR

**B.S. DEGREE**

**Prerequisites **PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261, with appropriate math coreqs; PHYS 205L, 206L, or PHYS 165L, 166L

**Number of courses **8 term courses beyond prereqs (incl senior req)

**Specific courses required **PHYS 301 or other advanced math course; PHYS 401, 402, and either APHY 439 or PHYS 440, in sequence

**Distribution of courses **3 advanced electives approved by DUS

**Senior requirement **PHYS 471 or 472 or equivalent

**B.S. DEGREE, INTENSIVE MAJOR**

**Prerequisites **PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261, with appropriate math coreqs; PHYS 205L, 206L, or PHYS 165L, 166L

**Number of courses** 10 term courses beyond prereqs (incl senior req)

**Specific courses required **PHYS 301 or other advanced math course; PHYS 410, 440, 441, 420, 430, in sequence; PHYS 382L

**Distribution of courses **1 advanced elective approved by DUS

### Courses

*** PHYS 050a / APHY 050a, Science of Modern Technology
** Daniel Prober

Examination of the science behind selected advances in modern technology. Focus on the scientific and contextual basis of each advance. Topics are developed by the participants with the instructor and with guest lecturers, and may include nanotechnology, quantum computation and cryptography, optical systems for communication and medical diagnostics, transistors, satellite imaging and global positioning systems, large-scale immunization, and DNA made to order. Enrollment limited to freshmen. Preregistration required; see under Freshman Seminar Program. SC RP

*** PHYS 100b / APHY 100b / ENAS 100b / EVST 100b / G&G 105b, Energy Technology and Society
** Daniel Prober, Michael Oristaglio, and Julie Paquette

The technology and use of energy. Impacts on the environment, climate, security, and economy. Application of scientific reasoning and quantitative analysis. Intended for non–science majors with strong backgrounds in math and science. Enrollment limited to 24. For application instructions, visit the course site on Classes*v2. QR, SC

*** PHYS 107a / MB&B 107a, Being Human in STEM
** Simon Mochrie

A collaboratively-designed, project-oriented course that seeks to examine, understand, and disseminate how diversity of gender, race, religion, sexuality, economic circumstances, etc. shape the STEM experience at Yale and nationally, and that seeks to formulate and implement solutions to issues that are identified. Study of relevant peer-reviewed literature and popular-press articles. Implementation of a questionnaire and interviews of STEM participants at Yale. Creation of role-play scenarios for provoking discussions and raising awareness. Design and implementation of group interventions. SO

*** PHYS 120b, Quantum Physics and Beyond
** John Harris

Current topics in modern physics, beginning with quantum physics and continuing through subatomic physics, special and general relativity, cosmology, astrophysics, and string theory. SC

**PHYS 151b / APHY 151a or b / ENAS 151a or b, Multivariable Calculus for Engineers
** Beth Anne Bennett

An introduction to multivariable calculus focusing on applications to engineering problems. Topics include vector-valued functions, vector analysis, partial differentiation, multiple integrals, vector calculus, and the theorems of Green, Stokes, and Gauss. Prerequisite: MATH 115 or equivalent. QR RP

**PHYS 165La and PHYS 166Lb, General Physics Laboratory
** Staff

A variety of individually self-contained experiments are roughly coordinated with the lectures in PHYS 170, 171, and 180, 181 and illustrate and develop physical principles covered in those lectures. SC ½ Course cr per term

*** PHYS 170a and PHYS 171b, University Physics for the Life Sciences
** Staff

An introduction to classical physics with special emphasis on applications drawn from the life sciences and medicine. Fall-term topics include vectors and kinematics, Newton's laws, momentum, energy, random walks, diffusion, fluid mechanics, mathematical modeling, and statistical mechanics. Spring-term topics include oscillations, waves, sound, electrostatics, circuits, Maxwell's equations, electromagnetic waves and optics, gene circuits, and quantum mechanics. Essential mathematics are introduced and explained as needed. Completion of MATH 112 or equivalent is prerequisite for PHYS 170. MATH 115 is recommended prior to or concurrently with PHYS 171. QR, SC

**PHYS 180a and PHYS 181b, University Physics
** Staff

A broad introduction to classical and modern physics for students who have some previous preparation in physics and mathematics. Fall-term topics include Newtonian mechanics, gravitation, waves, and thermodynamics. Spring-term topics include electromagnetism, optics, special relativity, and quantum physics. Concurrently with MATH 115 and 120 or equivalents. See comparison of introductory sequences and laboratories in the YCPS. May not be taken for credit after PHYS 170, 171. QR, SC

**PHYS 200a and PHYS 201b, Fundamentals of Physics
** Helen Caines and Francis Robinson

A thorough introduction to the principles and methods of physics for students who have good preparation in physics and mathematics. Emphasis on problem solving and quantitative reasoning. Fall-term topics include Newtonian mechanics, special relativity, gravitation, thermodynamics, and waves. Spring-term topics include electromagnetism, geometrical and physical optics, and elements of quantum mechanics. Prerequisite: MATH 115 or equivalent. See comparison of introductory sequences and laboratories in the YCPS. QR, SC

**PHYS 205La or b and PHYS 206La or b, Modern Physical Measurement
** Karsten Heeger and Staff

A two-term sequence of experiments in classical and modern physics for students who plan to major in Physics. In the first term, the basic principles of mechanics, electricity, and magnetism are illustrated in experiments designed to make use of computer data handling and teach error analysis. In the second term, students plan and carry out experiments illustrating aspects of wave and quantum phenomena and of atomic, solid state, and nuclear physics using modern instrumentation. May be begun in either term. SC ½ Course cr per term

*** PHYS 260a and PHYS 261b, Intensive Introductory Physics
** Charles Baltay

The major branches of physics—classical and relativistic dynamics, gravitation, electromagnetism, heat and thermodynamics, statistical mechanics, quantum physics—at a sophisticated level. For students majoring in the physical sciences, Mathematics, and Philosophy who have excellent training in and a flair for mathematical methods and quantitative analysis. Concurrently with MATH 230 and 231, or PHYS 301, or equivalent. QR, SC

**PHYS 295a / ASTR 255a, Research Methods in Astrophysics
** Marla Geha

The acquisition and analysis of astrophysical data, including the design and use of ground- and space-based telescopes, computational manipulation of digitized images and spectra, and confrontation of data with theoretical models. Examples taken from current research at Yale and elsewhere. Use of the Python programming language. A background in high school calculus and physics. No previous programming experience required. QR, SC RP

**PHYS 301a, Introduction to Mathematical Methods of Physics
** Oliver Baker

Topics include multivariable calculus, linear algebra, complex variables, vector calculus, and differential equations. Designed to give accelerated access to 400-level courses by providing, in one term, the essential background in mathematical methods. Recommended to be taken concurrently with PHYS 401 or 410. Prerequisite: PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261, or permission of instructor. QR

**PHYS 343b / ASTR 343b, Gravity, Astrophysics, and Cosmology
** Priyamvada Natarajan

Introduction to frontier areas of research in astrophysics and cosmology exploring ideas and methods. In-depth discussion of the physics underlying several recent discoveries including extrasolar planets—their discovery, properties, and issues of habitability; black holes—prediction of their properties from GR, observational signatures, and detection; and the accelerating universe—introduction to cosmological models and the discovery of dark energy. Prerequisites: PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261, or permission of instructor. QR, SC

**PHYS 344b, Quantum and Nanoscale Physics
** Sean Barrett

An introduction to cutting-edge developments in physics involving quantum information and/or nanotechnology. Background concepts in quantum mechanics, electromagnetism, and optics are introduced as necessary. Prerequisite: PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261, or permission of instructor. PHYS 301 or other advanced mathematics course recommended. QR, SC

*** PHYS 356b / ASTR 356b, Astrostatistics and Data Mining
** Hector Arce

Introduction to the statistical tools used to analyze and interpret astrophysical data, including common data mining techniques for finding patterns in large data sets and data-based prediction methods. Use of publicly available high-quality astronomical data from large surveys such as SDSS and 2MASS, and from space-based observatories such as Spitzer, Herschel, and WISE. Coding with the Python programming language. Prerequisite: ASTR 255 or equivalent. QR, SC

*** PHYS 382Lb, Advanced Physics Laboratory
** Steve Lamoreaux, Reina Maruyama, Nir Navon, and Sidney Cahn

Laboratory experiments with some discussion of theory and techniques. An advanced course focusing on modern experimental methods and concepts in atomic, optical, nuclear, and condensed matter physics. Intended to prepare students for independent research. For majors in the physical sciences. After or concurrently with PHYS 439 or 440, or with permission of instructor. PHYS 206L WR, SC

**PHYS 401a and PHYS 402b, Advanced Classical Physics from Newton to Einstein
** Ramamurti Shankar

Advanced physics as the field developed from the time of Newton to the age of Einstein. Topics include mechanics, electricity and magnetism, statistical physics, and thermodynamics. The development of classical physics into a "mature" scientific discipline, an idea that was subsequently shaken to the core by the revolutionary discoveries of quantum physics and relativity. Prerequisite: PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261. Concurrently with PHYS 301 or other advanced mathematics course. QR, SC

**PHYS 410a, Classical Mechanics
** Jack Harris

An advanced treatment of mechanics, with a focus on the methods of Lagrange and Hamilton. Lectures and problems address the mechanics of particles, systems of particles, and rigid bodies, as well as free and forced oscillations. Introduction to chaos and special relativity. Prerequisite: PHYS 170, 171, or 180, 181, or 200, 201, or 260, 261. Concurrently with PHYS 301 or other advanced mathematics course. QR, SC

*** PHYS 420a / APHY 420a, Thermodynamics and Statistical Mechanics
** A. Douglas Stone

An introduction to the laws of thermodynamics and their theoretical explanation by statistical mechanics. Applications to gases, solids, phase equilibrium, chemical equilibrium, and boson and fermion systems. PHYS 301 and 410 or equivalents QR, SC

**PHYS 428a / AMTH 428a / E&EB 428a / G&G 428a, Science of Complex Systems
** Jun Korenaga

Introduction to the quantitative analysis of systems with many degrees of freedom. Fundamental components in the science of complex systems, including how to simulate complex systems, how to analyze model behaviors, and how to validate models using observations. Topics include cellular automata, bifurcation theory, deterministic chaos, self-organized criticality, renormalization, and inverse theory. Prerequisite: PHYS 301, MATH 247, or equivalent. QR, SC

**PHYS 430b, Electromagnetic Fields and Optics
** David Poland

Electrostatics, magnetic fields of steady currents, electromagnetic waves, and relativistic dynamics. Provides a working knowledge of electrodynamics. Prerequisites: PHYS 301 and 410 or equivalents. QR, SC

**PHYS 439a / APHY 439a, Basic Quantum Mechanics
** Sohrab Ismail-Beigi

The basic concepts and techniques of quantum mechanics essential for solid-state physics and quantum electronics. Topics include the Schrödinger treatment of the harmonic oscillator, atoms and molecules and tunneling, matrix methods, and perturbation theory. Prerequisites: PHYS 181 or 201, PHYS 301, or equivalents, or permission of instructor. QR, SC

**PHYS 440b, Quantum Mechanics and Natural Phenomena I
** Witold Skiba

The first term of a two-term sequence covering principles of quantum mechanics with examples of applications to atomic physics. The solution of bound-state eigenvalue problems, free scattering states, barrier penetration, the hydrogen-atom problem, perturbation theory, transition amplitudes, scattering, and approximation techniques. Prerequisite: PHYS 410 or 401. QR, SC

**PHYS 441a, Quantum Mechanics and Natural Phenomena II
** Steve Lamoreaux

**PHYS 448a / APHY 448a, Solid-State Physics I
** Victor Henrich

The first term of a two-term sequence covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structure, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonances, phase transitions, dielectrics, magnetic materials, and superconductors. Prerequisites: APHY 322, 439, PHYS 420. QR, SC

**PHYS 449b / APHY 449b, Solid-State Physics II
** Michel Devoret

The second term of the sequence described under APHY 448. QR, SC

**PHYS 458a / APHY 458a, Principles of Optics with Applications
** Hui Cao

Introduction to the principles of optics and electromagnetic wave phenomena with applications to microscopy, optical fibers, laser spectroscopy, and nanostructure physics. Topics include propagation of light, reflection and refraction, guiding light, polarization, interference, diffraction, scattering, Fourier optics, and optical coherence. Prerequisite: PHYS 430. QR, SC

**PHYS 460a, Mathematical Methods of Physics
** Nicholas Read

Survey of mathematical techniques useful in physics. Physical examples illustrate vector and tensor analysis, group theory, complex analysis (residue calculus, method of steepest descent), differential equations and Green's functions, and selected advanced topics. Prerequisite: PHYS 301 or other advanced mathematics course. QR

*** PHYS 471a and PHYS 472b, Independent Projects in Physics
** Staff

Each student works on an independent project under the supervision of a member of the faculty or research staff. Students participate in a series of seminar meetings in which they present a talk on their project or research related to it. A written report is also submitted. For students with a strong background in Physics course work. RP