# Physics

35 Sloane Physics Laboratory, 203.432.3607

http://physics.yale.edu

M.S., M.Phil., Ph.D.

**Chair**

Paul Tipton

**Director of Graduate Studies**

Nikhil Padmanabhan [F] (graduatephysics@yale.edu)

Sean Barrett [Sp] (SPL 24, 203.432.6928, graduatephysics@yale.edu)

**Professors** Robert Adair (*Emeritus*), Charles Ahn (*Applied Physics*), Yoram Alhassid, Thomas Appelquist, Charles Bailyn (*Astronomy*), O. Keith Baker, Charles Baltay, Sean Barrett, Hui Cao (*Applied Physics*), Richard Casten (*Emeritus*), Flavio Cavanna (*Adjunct*), Paolo Coppi (*Astronomy*), David DeMille, Michel Devoret (*Applied Physics*), Frank Firk (*Emeritus*), Debra Fischer (*Astronomy*), Bonnie Fleming, Marla Geha (*Astronomy*), Steven Girvin, Leonid Glazman, Jack Harris, John Harris, Karsten Heeger, Jay Hirshfield (*Adjunct*), Jonathon Howard (*Molecular Biophysics & Biochemistry*), Francesco Iachello (*Emeritus*), Sohrab Ismaill-Beigi (*Applied Physics*), Steve Lamoreaux, Samuel MacDowell (*Emeritus*), Simon Mochrie, Vincent Moncrief, Priyamvada Natarajan (*Astronomy*), Corey O’Hern (*Mechanical Engineering & Materials Science*), Ornella Palamara (*Adjunct*), Peter Parker (*Emeritus*), Daniel Prober (*Applied Physics*), Nicholas Read, Jack Sandweiss (*Emeritus*), Peter Schiffer (*Applied Physics*), Robert Schoelkopf (*Applied Physics*), Ramamurti Shankar, Witold Skiba, Charles Sommerfield (*Emeritus*), A. Douglas Stone (*Applied Physics*), Hong Tang (*Electrical Engineering*), Paul Tipton, Thomas Ullrich (*Adjunct*), C. Megan Urry, Pieter van Dokkum (*Astronomy*), John Wettlaufer (*Geology & Geophysics*), Robert Wheeler (*Emeritus*), Werner Wolf (*Emeritus*), Michael Zeller (*Emeritus*)

**Associate Professors** Murat Acar (*Molecular, Cellular, & Developmental Biology*), Helen Caines, Damon Clark (*Molecular, Cellular, & Developmental Biology*), Sarah Demers, Thierry Emonet (*Molecular, Cellular, & Developmental Biology*), Walter Goldberger, Liang Jiang (*Applied Physics*), Reina Maruyama, Daisuke Nagai, Nikhil Padmanabhan, David Poland, Peter Rakich (*Applied Physics*)

**Assistant Professors** Eric Brown (*Mechanical Engineering & Materials Science*), Meng Cheng, Benjamin Machta, David Moore, John Murray (*Psychiatry*), Michael Murrell (*Biomedical Engineering*), Nir Navon, Laura Newburgh

## Fields of Study

Fields include atomic physics and quantum optics; nuclear physics; particle physics; astrophysics and cosmology; condensed matter; biological physics; quantum information physics; applied physics; and other areas in collaboration with the School of Engineering & Applied Science, and the departments of Applied Physics; Astronomy; Chemistry; Geology and Geophysics; Molecular Biophysics and Biochemistry; and Molecular, Cellular, and Developmental Biology.

## Special Admissions Requirements

The prerequisites for work toward a Ph.D. degree in physics include a sound undergraduate training in physics and a good mathematical background. The GRE General Test and the Subject Test in Physics are required.

## Integrated Graduate Program in Physical and Engineering Biology (PEB)

Students applying to the Ph.D. program in Physics may also apply to be part of the PEB program. See the description under Non-Degree-Granting Programs, Councils, and Research Institutes for course requirements, and http://peb.yale.edu for more information about the benefits of this program and application instructions.

## Special Requirements for the Ph.D. Degree

To complete the course requirements, students are expected to take a set of six term courses: five foundational courses and one elective. The five core courses (1. PHYS 500, Advanced Classical Mechanics; 2. PHYS 508, Quantum Mechanics I; 3. PHYS 502, Electromagnetic Theory I; 4. PHYS 512, Statistical Physics I; and 5. PHYS 608, Quantum Mechanics II) serve to complete the student’s undergraduate training in classical and quantum physics. For the sixth course, students select from the list of graduate elective courses offered by the Physics or Applied Physics departments, or courses offered by other departments with the approval of the DGS. In addition, all students are required to engage in a research project by taking PHYS 990, Special Investigations. In their first year of study, students must take, at a minimum, the foundational courses one through four, along with the research seminar courses: PHYS 515, Topics in Modern Physics Research, and PHYS 590, Responsible Conduct in Research for Physical Scientists. Certain equivalent course work or successful completion of a pass-out examination may allow substitution of elective courses for individual students.

Students who have completed their course requirements with satisfactory grades, passed the qualifying examination, and submitted an acceptable thesis prospectus are recommended for admission to candidacy. (A grade of Honors in PHYS 990, Special Investigations, may be counted toward the Graduate School requirement of two grades of Honors.) The qualifying examination, normally taken at the beginning of the third term (and no later than the beginning of the fifth term), consists of four separate, written exams on Classical Mechanics, Electromagnetic Theory, Statistical Mechanics, and Quantum Mechanics. Students normally submit the dissertation prospectus before the end of the third year of study.

There is no foreign language requirement. Teaching experience is regarded as an integral part of the graduate training program. During their studies, students are expected to serve four terms as teaching fellows at the TF-10 level, usually in the first two years. Formal association with a dissertation adviser normally begins in the fourth term, after the qualifying examination has been passed and required course work has been completed. An adviser from a department other than Physics can be chosen in consultation with the director of graduate studies (DGS), provided the dissertation topic is deemed suitable for a physics Ph.D.

## Master’s Degrees

**M.Phil.** Students who have successfully advanced to candidacy qualify for the M.Phil. degree.

**M.S. (en route to the Ph.D.)** Students who complete all courses numbered one through four above, plus one of the following: PHYS 608, Quantum Mechanics II; PHYS 990, Special Investigations; or an advanced elective (all with a satisfactory record) qualify for the M.S. degree. Certain equivalent course work or successful completion of a pass-out examination may allow individual students to substitute an elective course for a required one.

Program materials are available upon request to the Director of Graduate Studies, Department of Physics, Yale University, PO Box 208120, New Haven CT 06520-8120; e-mail, graduatephysics@yale.edu; website, http://physics.yale.edu.

## Courses

**PHYS 500a, Advanced Classical Mechanics** Yoram Alhassid

Newtonian dynamics, Lagrangian dynamics, and Hamiltonian dynamics. Rigid bodies and Euler equations. Oscillations and eigenvalue equations. Classical chaos. Introduction to dynamics of continuous systems.

TTh 10:45am-12pm

**PHYS 502b, Electromagnetic Theory I** A. Douglas Stone

Classical electromagnetic theory including boundary-value problems and applications of Maxwell equations. Macroscopic description of electric and magnetic materials. Wave propagation.

TTh 9am-10:15am

**PHYS 504b, Modern Physics Measurements** Nir Navon and Steve Lamoreaux

A laboratory course with experiments and data analysis in soft and hard condensed matter, nuclear and elementary particle physics.

TTh 1:30pm-4:30pm

**PHYS 506a, Mathematical Methods of Physics** Nicholas Read

Survey of mathematical techniques useful in physics. Includes vector and tensor analysis, group theory, complex analysis (residue calculus, method of steepest descent), differential equations and Green’s functions, and selected advanced topics.

MW 9am-10:15am

**PHYS 508a, Quantum Mechanics I** Ramamurti Shankar

The principles of quantum mechanics with application to simple systems. Canonical formalism, solutions of Schrödinger’s equation, angular momentum, and spin.

MW 11:35am-12:50pm

**PHYS 512b, Statistical Physics I** Meng Cheng

Review of thermodynamics, the fundamental principles of classical and quantum statistical mechanics, canonical and grand canonical ensembles, identical particles, Bose and Fermi statistics, phase transitions and critical phenomena, enormalization group, irreversible processes, fluctuations.

TTh 2:30pm-3:45pm

**PHYS 515a, Topics in Modern Physics Research** Yoram Alhassid

A seminar course intended to provide an introduction to current research in physics and an overview of physics research opportunities at Yale.

F 10:40am-11:55am

**PHYS 523b / CB&B 523b / ENAS 541b / MB&B 523b, Biological Physics** Simon Mochrie

The course has two aims: (1) to introduce students to the physics of biological systems and (2) to introduce students to the basics of scientific computing. The course focuses on studies of a broad range of biophysical phenomena including diffusion, polymer statistics, protein folding, macromolecular crowding, cell motion, and tissue development using computational tools and methods. Intensive tutorials are provided for MATLAB including basic syntax, arrays, for-loops, conditional statements, functions, plotting, and importing and exporting data.

TTh 1pm-2:15pm

**PHYS 524a, Introduction to Nuclear Physics** Bonnie Fleming and Karsten Heeger

Introduction to a wide variety of topics in nuclear in nuclear physics. A number of related nuclear models as well as experimental methods are discussed. The course also covers topics in weak interactions, neutrino physics, neutrinoless double beta decay, and relativistic heavy ion collisions. The aim is to give a broad perspective on the subject and to develop the key ideas in simple ways, with more weight on physics ideas than on mathematical formalism. The course assumes no prior knowledge of nuclear physics and only elementary quantum mechanics. It is accessible to advanced undergraduates.

TTh 9am-10:15am

**PHYS 526b, Introduction to Elementary Particle Physics** Oliver Baker

An overview of particle physics, including an introduction to the standard model, experimental techniques, symmetries, conservation laws, the quark-parton model, and open questions in particle physics.

MW 1pm-2:15pm

**PHYS 528b / ENAS 848b, Soft Condensed Matter Physics** Staff

An introduction to the physics and phenomenology of soft condensed matter: classical systems with mesoscale structure where thermal fluctuations and interfacial forces play essential roles. Discussion of applications to materials science/engineering, nanotechnology, and molecular/cellular biology. Essential concepts from statistical thermodynamics, classical mechanics, and electricity and magnetism are reviewed/developed as needed.

MW 11:35am-12:50pm

**PHYS 538b, Introduction to Relativistic Astrophysics and General Relativity** Walter Goldberger

Basic concepts of differential geometry (manifolds, metrics, connections, geodesics, curvature); Einstein’s equations and their application to such areas as cosmology, gravitational waves, black holes.

MW 2:30pm-3:45pm

**PHYS 548a, Solid State Physics I** Sohrab Ismail-Beigi

A two-term sequence (with PHYS 549) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.

TTh 1pm-2:15pm

**PHYS 549b, Solid State Physics II** Vidvuds Ozolins

A two-term sequence (with PHYS 548) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.

TTh 1pm-2:15pm

**PHYS 561a / CB&B 561a / MB&B 561a / MBIO 561a / MCDB 561a, Introduction to Dynamical Systems in Biology** Damon Clark, Kathryn Miller-Jensen, and Jonathon Howard

Study of the analytic and computational skills needed to model genetic networks and protein signaling pathways. Review of basic biochemical concepts including chemical reactions, ligand binding to receptors, cooperativity, and Michaelis-Menten enzyme kinetics. Deep exploration of biological systems including: kinetics of RNA and protein synthesis and degradation; transcription activators and repressors; lyosogeny/lysis switch of lambda phage and the roles of cooperativity and feedback; network motifs such as feed-forward networks and how they shape response dynamics; cell signaling, MAP kinase networks and cell fate decisions; bacterial chemotaxis; and noise in gene expression and phenotypic variability. Students learn to model using MATLAB in a series of in-class hackathons that illustrate biological examples discussed in lectures.

TTh 2:30pm-3:45pm

**PHYS 562b / AMTH 765b / CB&B 562b / INP 562b / MB&B 562b / MCDB 562b, Dynamical Systems in Biology** Thierry Emonet and Jonathon Howard

This course covers advanced topics in computational biology. How do cells compute, how do they count and tell time, how do they oscillate and generate spatial patterns? Topics include time-dependent dynamics in regulatory, signal-transduction, and neuronal networks; fluctuations, growth, and form; mechanics of cell shape and motion; spatially heterogeneous processes; diffusion. This year, the course spends roughly half its time on mechanical systems at the cellular and tissue level, and half on models of neurons and neural systems in computational neuroscience. Prerequisite: MCDB 561 or equivalent, or a 200-level biology course, or permission of the instructor.

TTh 2:30pm-3:45pm

**PHYS 590b / APHY 590b, Responsible Conduct in Research for Physical Scientists** Staff

Required seminar for all first-year students.

HTBA

**PHYS 608b, Quantum Mechanics II** Nicholas Read

Approximation methods, scattering theory, and the role of symmetries. Relativistic wave equations. Second quantized treatment of identical particles. Elementary introduction to quantized fields.

MW 11:35am-12:50pm

**PHYS 609a, Relativistic Field Theory I** Thomas Appelquist

The fundamental principles of quantum field theory. Interacting theories and the Feynman graph expansion. Quantum electrodynamics including lowest order processes, one-loop corrections, and the elements of renormalization theory.

MW 11:35am-12:50pm

**PHYS 624b, Group Theory** Witold Skiba

Lie algebras, Lie groups, and some of their applications. Representation theory. Explicit construction of finite-dimensional irreducible representations. Invariant operators and their eigenvalues. Tensor operators and enveloping algebras. Boson and fermion realizations. Differential realizations. Quantum dynamical applications.

MW 9am-10:15am

**PHYS 628a / APHY 628a, Statistical Physics II** Meng Cheng

An advanced course in statistical mechanics. Topics may include mean field theory of and fluctuations at continuous phase transitions; critical phenomena, scaling, and introduction to the renormalization group ideas; topological phase transitions; dynamic correlation functions and linear response theory; quantum phase transitions; superfluid and superconducting phase transitions; cooperative phenomena in low-dimensional systems.

TTh 2:30pm-3:45pm

**PHYS 630b, Relativistic Field Theory II** Thomas Appelquist

An introduction to non-Abelian gauge field theories, spontaneous symmetry breakdown, and unified theories of weak and electromagnetic interactions. Renormalization group methods, quantum chromodynamics, and nonperturbative approaches to quantum field theory.

TTh 1pm-2:15pm

**PHYS 632a, Quantum Many-Body Theory II** Leonid Glazman

A second course in quantum many-body theory, covering the core physics of electron systems, with emphasis on the electron-electron interaction, on the role of dimensionality, on the coupling either to magnetic impurities leading to the well-known Kondo effect or to the electromagnetic noise. Applications to mesoscopic systems and cold atomic gases are also developed.

TTh 4pm-5pm

**PHYS 669a, Relativistic Field Theory III** Walter Goldberger

This course focuses on applications of quantum field theory to phenomena in particle physics and gravity. The first part consists of a detailed discussion of the Standard Model, both its formal properties and experimental predictions. The second part is a survey of modern scattering amplitude methods in gauge theory (with applications to collider physics) and in quantum gravity. The last part discusses the applications of field theory techniques to gravitational wave sources, including a brief introduction to LIGO phenomenology.

MW 11:35am-12:50pm

**PHYS 675a / APHY 675a, 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, nanophotonics, plasmonics, and metamaterials. Topics include propagation of light, reflection and refraction, guiding light, polarization, interference, diffraction, scattering, Fourier optics, and optical coherence.

TTh 11:35am-12:50pm

**PHYS 677a / APHY 677a, Noise, Dissipation, Amplification, and Information** Michel Devoret

Graduate-level non-equilibrium statistical physics applied to noise phenomena, both classical and quantum. The aim of the course is to explain the fundamental link between the random fluctuations of a physical system in steady state and the response of the same system to an external perturbation. Several key examples in which noise appears as a resource rather than a limitation are treated: spin relaxation in nuclear magnetic resonance (motional narrowing), Johnson-Nyquist noise in solid state transport physics (noise thermometry), photon correlation measurements in quantum optics (Hanbury Brown-Twiss experiment), and so on. The course explores both passive and active systems. It discusses the ultimate limits of amplifier sensitivity and speed in physics measurements.

MW 9am-10:15am

**PHYS 762a or b / CHEM 562La or b, Laboratory in Instrument Design and the Mechanical Arts** Kurt Zilm and David Johnson

Familiarization with modern machine shop practices and techniques. Use of basic metalworking machinery and instruction in techniques of precision measurement and properties of commonly used metals, alloys, and plastics.

HTBA

**PHYS 816b / APHY 816b, Techniques Microwave Measurement** Robert Schoelkopf

An advanced course covering the concepts and techniques of radio-frequency design and their application in making microwave measurements. The course begins with a review of lumped element and transmission line circuits, network analysis, and design of passive elements, including filters and impedance transformers. We continue with a treatment of passive and active components such as couplers, circulators, amplifiers, and modulators. Finally, we employ this understanding for the design of microwave measurement systems and techniques for modulation and signal recovery, to analyze the performance of heterodyne/homodyne receivers and radiometers.

TTh 2:30pm-3:45pm

**PHYS 990a, Special Investigations** Staff

Directed research by arrangement with individual faculty members and approved by the DGS.

HTBA

**PHYS 991a / ENAS 991a / MB&B 591a / MCDB 591a, Integrated Workshop** Corey O'Hern, Mark Gerstein, Scott Holley, Marcus Bosenberg, Madhusudhan Venkadesan, Michael Murrell, and Nikhil Malvankar

This required course for students in the PEB graduate program involves a series of modules, co-taught by faculty, in which students from different academic backgrounds and research skills collaborate on projects at the interface of physics, engineering, and biology. The modules cover a broad range of PEB research areas and skills. The course starts with an introduction to Matlab, since Matlab is used throughout the course for analysis, simulations, and modeling. ½ Course cr

MW 9am-10:15am