# Undergraduate Program

#### Contact Information

Interim Director of Undergraduate Studies:

Professor Daniel Zwanziger

E-mail: daniel.zwanziger@nyu.edu

Phone: 212-998-7732

Undergraduate Advisor:

Professor Daniel Zwanziger

E-mail: daniel.zwanziger@nyu.edu

Phone: 212-998-7732

Assistant to the Director of Undergraduate Studies:

Bill LePage

E-mail: wlp1@nyu.edu

Phone: 212-998-7704

Physics is the most basic of the natural sciences. It is concerned with understanding the world on all scales of length, time, and energy. The methods of physics are diverse, but they share a common objective to develop and refine fundamental models that quantitatively explain observations and the results of experiments. The discoveries of physics, exemplified by the laws of physics, rank among the most important achievements of human inquiry, and have had an enormous impact on human culture and civilization. Members of the department carry out research in the fields of astrophysics, biophysics, cosmology, elementary particle physics, gravitation, hard and soft condensed matter physics, and statistical physics. Experimental work is carried out in state-of-the art laboratories in the department and at national and international facilities such as the Large Hadron Collider at CERN and astronomical telescopes in space. The majority of NYU physics graduates go to graduate school in physics, and the rest pursue careers in medicine, finance, engineering and law. 27 percent of majors are women; 5% black, 16% Asian, and 8% Latino.

The NYU SPS chapter organizes a mentoring program that pairs first-year students with upper division Physics Majors, organizes lunches for students, and sponsors weekly sessions where faculty are invited to discuss their research. There are many opportunities for faculty-student interaction. These include the weekly colloquium and weekly events such as the CCPP (Center for Cosmology and Particle Physics) Brown Bag, in which faculty present talks on their research.

The educational programs of the Department are aimed at providing a range of courses to meet the needs of different student groups. For undergraduate physics majors, there is a rigorous core program, exposure to current frontiers, and opportunities for research. For science majors outside physics there are technical courses that emphasize the fundamental physical laws that underpin other sciences; and for non-science majors there are non-technical courses that introduce some of the most important concepts of physics and their impact on the contemporary world.

#### Faculty

**Professors Emeriti:**

Bederson, Borowitz, Brown, Glassgold, Hoffert, Hohenberg, Levy, Lowenstein, Richardson, Robinson, Rosenberg, Sculli, Yarmus

**Silver Professors; Professors of Physics:**

Chaikin, Dvali, Pine

**Collegiate Professors; Professors of Physics:**

Farrar, Mincer

**Professors:**

Budick, Gabadadze, Grier, Grosberg, Hogg, Kent, Nemethy, Percus, Porrati, Scoccimarro, Sokal, Stein, Stroke, Weiner, Zhang, Zwanziger

**Associate Professors:**

Brujic, Blanton, Cranmer, Gruzinov, Kleban, MacFadyen, Mitra, Sleator

**Assistant Professors:**

Dubovsky, Gershow, Haas, Modjaz, Ruderman, Tinker, Wray, Zidovska

**Clinical Professor:**

Adler

##### Program

#### DEPARTMENTAL OBJECTIVES

**Majors:** The Major programs are designed to meet a number of goals: They provide good preparation for graduate school; indeed many of the Majors go on to some of the world's best graduate programs. They develop a range of technical skills, most of which relate to the challenging intellectual problems of building quantitative theoretical models and making precise measurements of physically interesting phenomena. The Major programs are also designed to satisfy curiosity about the fundamental laws that govern every aspect of the world from the interactions of subatomic particles to the origin and behavior of the entire Universe.

The Major programs are simultaneously very deep and very broad. Course work includes both theoretical subjects and experimental activity in laboratories. The programs are designed to give students flexibility in Years 3 and 4 to pursue interdisciplinary activities, spend time abroad, or go into greater depth in a subject or into original research.

The Department is a collegial place where the faculty and the students get to know one another well; there are regular formal and informal seminars; there is a thriving Society of Physics Students; and students and faculty often collaborate on original research problems. Many of the Majors participate in original research and co-author scientific publications. For all of these reasons, and in addition to the rigor of the courses, the Majors are extremely well prepared for a wide range of activities—not just in scientific research, but also in professional and engineering pursuits—or any area where abstract thinking and quantitative modeling of real systems is necessary and rewarded.

**Non-majors:** For non-science majors, there are non-technical courses that introduce some of the concepts and events that are most important to understanding physics and its impact on the contemporary world. For science majors outside of physics, there are technical courses that bring breadth or ideas about fundamental laws that underpin the other sciences. The Department provides courses designed to meet the pre-professional goals of pre-health students and students in engineering disciplines.

**Minors:** There are Minor programs in Physics and Astrophysics for students who are interested in obtaining significant experience in the ideas of physics without committing to the Major or without obtaining a comprehensive mathematical background.

#### SUGGESTED PROGRAMS FOR MAJORS IN PHYSICS

#### B.A. PROGRAM

**Bachelor of Arts in Physics:** The Major in Physics consists of the following courses:

*Year 1:* MATH-UA 121, MATH-UA 122, PHYS-UA 91, PHYS-UA 71, PHYS-UA 93, and PHYS-UA 72

*Year 2:* MATH-UA 123, PHYS-UA 95, PHYS-UA 73, PHYS-UA 105, PHYS-UA 106, and PHYS-UA 74

*Years 3 and 4:* PHYS-UA 112, PHYS-UA 123, PHYS-UA 131, PHYS-UA 140, and two electives from among the advanced Physics courses

**Mathematics:** The calculus requirement may be satisfied by taking Honors Calculus I, II (MATH-UA 221,0222) or Calculus I, II, III (MATH-UA 121-123). Students who take the Honors Calculus sequence begin it in the fall semester of their freshman year. Students who complete Honors Calculus I, II are encouraged to take Linear Algebra (MATH-UA 140) in the fall term of the second year. Variations may be constructed with the approval of the director of undergraduate studies. In addition, Students are advised to take advanced mathematics courses as they proceed in the Major.

**Double major including physics:** The Major offers flexibility to complete the requirements for a second major in the College. Students may wish to combine a major in physics with a major in a field such as mathematics, computer science, chemistry, economics, or biology. Students should consult the director of undergraduate studies in their freshman year to outline a program that is best tailored to their needs.

#### B.S. PROGRAM

#### Bachelor of Science in Physics:

The B.S. degree involves breadth in the sciences in addition to the Physics Major. The B.S. degree in physics will be granted to students completing the following:- The required courses for the B.A. Major, including one of the Physics electives
- Computational Physics (PHYS-UA 210)
- Two courses in chemistry at or above the level of General Chemistry I, II (CHEM-UA 101,102)
- A course in biology at or above the level of Principles of Biology (BIOL-UA 11) or in chemistry above the level of General Chemistry II (CHEM-UA 102)

#### MINORS

**Minor in Physics:** Provides the student with a general survey of the field, plus specialized study. Consists of four of the following courses, or three of the following courses plus one of the courses listed under the minor in astronomy: PHYS-UA 10, PHYS-UA 11, PHYS-UA 12, PHYS-UA 20, and all courses numbered above and including PHYS-UA 91 (except for pure laboratory courses).

**Minor in Astronomy:** Provides a comprehensive introduction to astronomy, including modern concepts, historical ideas, and observational experience. Consists of four courses; PHYS-UA 07 is required, plus the three following courses (or two of the following, and one of the courses listed under the minor in physics): PHYS-UA 08, PHYS-UA 13, PHYS-UA 14, and PHYS-UA 150.

#### HONORS PROGRAM

Candidates for a degree with honors in physics must complete the requirements for the B.A. Major described above. They must also complete the equivalent of a semester of experimental or theoretical research. Students who wish to fulfill this requirement should discuss possible options, such as independent study courses, with the director of undergraduate studies. A research paper based on this work must be prepared and orally presented. For additional general requirements for a degree with honors, please see the Honors and Awards section of this bulletin.

##### Courses

The following courses are lectures unless otherwise indicated.

**The
Universe: Its Nature and History
**PHYS-UA
7

*Offered every year. 4 points.*

Qualitative introduction to our understanding of the nature and evolution of the universe. Topics include the creation of the cosmos; its explosive evolution, present structure, and ultimate fate; the nature of stars and galaxies; the structure and evolution of our Milky Way; the birth, life, and eventual death of the solar system; our place and role in the universe; and the relationship of modern astronomical ideas to other cultural disciplines.

**Origins
of Astronomy
**PHYS-UA
8

*Identical to MEDI-UA 8. Offered every other year. 4 points.*

Introduction to the historical development of astronomy, from earliest times through the Copernican revolution. Traces the changes in our perception of the heavens and the influences that led to those changes, from astrology to the discoveries of Galileo and the physics of Newton. Includes descriptive astronomy of the solar system and a trip to the Hayden Planetarium.

**Sound
and Music
**PHYS-UA
10

*Assumes high school-level mathematics background. Offered every year. 4 points.*

Explores the production of musical sound and how it is perceived by us, dealing mainly with the physical basis of sound. Covers sound waves, resonance, how musical instruments produce sound, the concepts of scales and harmony, physical acoustics, physiological factors of perception, acoustics of auditoria, and sound recording and reproduction. Develops the necessary physics for the course, as needed.

**General
Physics I
**PHYS-UA
11

*Prerequisite: Calculus I (MATH-UA 121) with a minimum grade of C or equivalent, or completion of the Mathematics for Economics I and II sequence (MATH-UA 211 and 212), or permission of the instructor. Lecture, laboratory, and recitation. Not open to students who have completed Physics I (PHYS-UA 91) with a grade of C-minus or better. Offered in the fall. 5 points.*

Begins a two-semester introduction to physics intended primarily for preprofessional students and for those majoring in a science other than physics, although well-prepared students may wish to take the Physics I, II, III three-semester sequence for majors (with corequisite laboratories), below. Topics include kinematics and dynamics of particles; momentum, work, and energy; gravitation; circular, angular, and harmonic motion; mechanical and thermal properties of solids, liquids, and gases.

**General
Physics II
**PHYS-UA
12

*Prerequisite: General Physics I (PHYS-UA 11) with a grade of C-minus or better or permission of the department. Lecture, laboratory, and recitation. Offered in the spring. 5 points.*

Continuation of General Physics I (PHYS-UA 11). Topics include electric charge, field, and potential; magnetic forces and fields; resistive, capacitive, and inductive circuits; electromagnetic induction; wave motion; electromagnetic waves; geometrical optics; interference, diffraction, and polarization of light; relativity; atomic and nuclear structure; elementary particle physics.

**Observational
Astronomy
**PHYS-UA
13

*Prerequisite: The Universe: Its Nature and History (PHYS-UA 7) or higher, or permission of the instructor for nonscience majors and minors; no prerequisite for science majors and minors or those who have satisfied the Core Natural Science I requirement. Lecture and laboratory. Offered every year. 4 points.*

Introduction to the theory and practice of technical amateur astronomy. The approach is hands-on, with weekly evening laboratory/observing sessions. Topics include astronomical coordinate systems, optics, how to use a telescope, and the phenomena that can be seen in the urban night sky. Observing sessions involve the use of eight-inch telescopes.

**Physics and Astronomy in the Renaissance
**PHYS-UA
14

*No prerequisites. Lecture. Typically offered in the spring. 4 points.*

In addition to the magnificent flowering of the arts in the Renaissance, the period was also one of extraordinary advances in science, in particular in astronomy and physics. We examine this advance, emanating from scientific developments in European and Italian centers of learning during the Renaissance and at the start of the Age of Enlightenment, in the light of prior wisdom. We begin with a discussion of the true or mistaken views of the ancient Greeks and their transmission through Islamic centers of learning, but primarily focus on the "Copernican Revolution" of Nicolas Copernicus, Tycho Brahe, Johannes Kepler, and Galileo Galilei that was the beginning of observational science and astronomy. Also included are the truly universal scientist, engineer, and artist Leonardo da Vinci; the world’s first cosmologist, Giordano Bruno; and Sir Isaac Newton, whose Laws of Mechanics and Law of Universal Gravitation were the crowning culmination of the "Scientific Revolution."

**20th-Century
Concepts of Space, Time, and Matter
**PHYS-UA
20

*Assumes high school-level geometry and intermediate algebra background. Not open to students who have completed Natural Science I: Einstein's Universe (CORE-UA 204). Offered every year. 4 points.*

The 20th century witnessed two major revolutions in man's concepts of space, time, and matter. Einstein's special and general theories of relativity: implications of the special theory for our understanding of the unity of space and time, and of the general theory for our understanding of the nature of gravity. Quantum mechanics: a new picture of the basic structure and interactions of atoms, molecules, and nuclei. Topics include the uncertainty principle, wave-particle duality, and the continuing search for the fundamental constituents of matter.

**Introductory
Experimental Physics I
**PHYS-UA
71

*Typically taken with Physics I (PHYS-UA 91). Offered in the fall. 2 points.*

Introduces essential experimental techniques, including setup and operation of basic laboratory equipment, elementary experimental design, statistics and inference, and computational data analysis. Experimental techniques are introduced in the context of classic physics experiments.

**Introductory
Experimental Physics II
**PHYS-UA
72

*Prerequisite: Introductory Experimental Physics I (PHYS-UA 71). Typically taken with Physics II (PHYS-UA 93). Offered in the spring. 2 points.*

Continuation of Introductory Experimental Physics I (PHYS-UA 71).

**Intermediate
Experimental Physics I
**PHYS-UA
73

*Prerequisite: Introductory Experimental Physics II (PHYS-UA 72). Typically taken with Physics III (PHYS-UA 95). Offered in the fall. 2 points.*

Develops further the experimental techniques introduced in Introductory Experimental Physics I, II (PHYS-UA 71, 72) in the context of more advanced experiments.

**Intermediate
Experimental Physics II
**PHYS-UA
74

*Prerequisite: Intermediate Experimental Physics I (PHYS-UA 73). Typically taken with Classical and Quantum Waves (PHYS-UA 105). Offered in the spring. 2 points.*

Continuation of Intermediate Experimental Physics I (PHYS-UA 73).

**Physics
I
**PHYS-UA
91

*Corequisite: Calculus I (MATH-UA 121) or Honors Calculus I: Accelerated Calculus with Linear Algebra (MATH-UA 221). Physics majors must also register for Introductory Experimental Physics I (PHYS-UA 71). Lecture and recitation. Offered in the fall. 3 points.*

With PHYS-UA 93 and PHYS-UA 95, forms a three-semester sequence that must be taken in order, starting in the fall semester. Intended for physics majors and other interested science and mathematics majors. Topics include kinematics and dynamics of particles; energy and momentum; rotational kinematics and dynamics; harmonic oscillators; gravitational fields and potentials; special relativity.

**Physics
II
**PHYS-UA
93

*Prerequisite: Physics I (PHYS-UA 91) with a grade of C or better, or permission of the department. Corequisite: Calculus II (MATH-UA 122) or Honors Calculus II: Accelerated Calculus with Linear Algebra (MATH-UA 222). Physics majors must also register for Introductory Experimental Physics II (PHYS-UA 72). Lecture and recitation. Offered in the spring. 3 points.*

Continuation of Physics I (PHYS-UA 91). Topics include electrostatics; dielectrics; currents and circuits; the magnetic field and magnetic materials; induction; AC circuits; Maxwell's equations.

**Physics
III
**PHYS-UA
95

*Prerequisite: Physics II (PHYS-UA 93) with a grade of C or better, or permission of the department. Corequisite: Calculus III (MATH-UA 123) or Honors Calculus II: Accelerated Calculus with Linear Algebra (MATH-UA 222). Physics majors must also register for Intermediate Experimental Physics I (PHYS-UA 73). Lecture and recitation. Offered in the fall. 3 points.*

Continuation of Physics II (PHYS-UA 93). Topics include wave motion; Fourier series; sound; the reflection, refraction, interference, and diffraction of light; polarization; thermodynamics; kinetic theory and statistical physics.

**Classical
and Quantum Waves
**PHYS-UA
105

*Prerequisite: Physics III (PHYS-UA 95), and either Calculus III (MATH-UA 123) or Honors Calculus II: Accelerated Calculus with Linear Algebra (MATH-UA 222). Physics majors must also register for Intermediate Experimental Physics II (PHYS-UA 74). Lecture and recitation. Offered in the spring. 3 points.*

Topics include linear and nonlinear oscillators, resonance, coupled oscillators, normal modes, mechanical waves, light, matter waves, Fourier analysis, Fourier optics (diffraction), and an introduction to numerical (computer) methods for solving differential equations.

**Mathematical
Physics
**

*PHYS-UA 106 Prerequisite: Physics III (PHYS-UA 95). Lecture and recitation. Offered in the spring. 3 points.*

Mathematical preparation for the junior and senior courses in physics. Vector analysis, Fourier series and integrals, ordinary differential equations, matrices, partial differential equations, and boundary-value problems.

**Electronics
for Scientists
**PHYS-UA
110

*Identical to BIOL-UA 110, CHEM-UA 671. Prerequisite: General Physics II (PHYS-UA 12) or Physics II (PHYS-UA 93) or permission of the instructor. Lecture and laboratory. Offered in the fall. 5 points.*

Introduction to basic analog and digital electronics used in modern experiments and computers, for students from any science discipline. Basic concepts and devices presented in lecture are studied in the laboratory. Topics include filters, power supplies, transistors, operational amplifiers, digital logic gates, and both combinatorial and sequential digital circuits. Students learn the functions of modern electronic instrumentation and measurement.

**Advanced
Experimental Physics
**PHYS-UA
112

*Prerequisites: Intermediate Experimental Physics I, II (PHYS-UA 73, 74) and Quantum Mechanics I (PHYS-UA 123), or permission of the instructor. Laboratory. Offered every year. 3 points.*

Introduces the experiments and techniques of modern physics. Students choose their experiments and may use microcomputers for data analysis. Experimental areas include optical spectroscopy, the Mössbauer effect, cosmic rays, magnetic resonance, condensed matter, and relativistic mass.

**Dynamics
**PHYS-UA
120

*Prerequisites: Physics III (PHYS-UA 95) and Mathematical Physics (PHYS-UA 106). Offered every year. 3 points.*

Intermediate-level course on the principles and applications of dynamics. Emphasis on the formulation of problems and their numerical solution. Topics include conservation laws, central force motion, Lagrange's and Hamilton's equations, normal modes and small oscillations, and accelerated reference frames.

**Quantum
Mechanics I
**PHYS-UA
123

*Prerequisite: Classical and Quantum Waves (PHYS-UA 105). Offered every year. 3 points.*

Introduction to the experimental basis and formal mathematical structure of quantum mechanics. Topics include foundational experiments, wave-particle duality, wave functions, the uncertainty principle, the time-independent Schrödinger equation and its applications to one-dimensional problems and the hydrogen atom, angular momentum, and spin; Hilbert Space, operators, and observables; time-independent perturbation theory; atomic spectra.

**Quantum
Mechanics II
**PHYS-UA
124

*Prerequisite: Quantum Mechanics I (PHYS-UA 123). Offered every year. 3 points.*

Continuation of Quantum Mechanics I (PHYS-UA 123). Topics include the time-dependent Schrödinger equation, the Schrödinger and Heisenberg description of quantum systems, time-dependent perturbation theory, scattering theory, quantum statistics, and applications to atomic, molecular, nuclear, and elementary particle physics.

**Electricity
and Magnetism I
**PHYS-UA
131

*Prerequisites: Classical and Quantum Waves (PHYS-UA 105) and Mathematical Physics (PHYS-UA 106). Offered every year. 3 points.*

Introduction to electrodynamics with applications to physical problems. Topics include electrostatics, magnetostatics, Maxwell's equations, electromagnetic forces, electromagnetic waves, radiation from accelerating charges and currents, and special relativity.

**Optics
**PHYS-UA
133

*Prerequisite: Classical and Quantum Waves (PHYS-UA 105) or permission of the instructor. 3 points.*

Introduction to physical and geometrical optics. Wave phenomena including diffraction, interference, first-order and higher-order coherence. Holography, phase contrast and atomic force microscopy, and limits of resolution are some of the subjects included. Topics include atomic energy levels and radiative transitions, and detectors from photon counting to bolometers for infrared radiation.

**Condensed
Matter Physics
**PHYS-UA
135

*Prerequisite: Classical and Quantum Waves (PHYS-UA 105) or permission of the instructor. Offered every other year. 3 points.*

Designed as an introduction to condensed matter physics for students with knowledge of elementary quantum mechanics. Topics include crystal structure, lattice vibrations, and the energy band theory of metals and semiconductors; the electronic, magnetic, and optical properties of solids; and some modern research topics, such as the physics of nanostructures, soft condensed matter physics, and superconductivity.

**Readings
in Particle Physics
**PHYS-UA
136

*Prerequisite: Classical and Quantum Waves (PHYS-UA 105). Offered every other year. 3 points.*

The fundamental constituents of matter and the forces between them are microscopic, but also connect to the large-scale realms of astrophysics and cosmology. Close reading of journal articles in which the most important advances in elementary particle physics were first published, with overview lectures, discussion, and student presentations. Topics include the discovery of elementary particles in cosmic rays, antimatter, symmetries found in nature, and the invention of the Quark model of elementary particles and its experimental verification.

**Quantum Information and Quantum Computing
**PHYS-UA
138

*Prerequisite: Quantum Mechanics I (PHYS-UA 123). Offered every two years. 4 points.*

Quantum mechanical systems can be thought of as information-storing, information-processing, and information-transmitting systems. The theory of quantum information contains many surprising and counter-intuitive results, and can potentially have a significant impact on our lives and society. Topics include density operators, quantum communication, teleportation, quantum cryptography, entanglement and the Bell Inequalities, quantum computing, quantum algorithms, quantum error correction, quantum circuits, and experimental developments.

**Thermal
and Statistical Physics
**PHYS-UA
140

*Prerequisites: Classical and Quantum Waves (PHYS-UA 105) and Mathematical Physics (PHYS-UA 106). Offered every year. 3 points.*

Topics include relation of entropy to probability and energy to temperature; the laws of thermodynamics; Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics; equations of state for simple gases and chemical and magnetic systems; and elementary theory of phase transitions.

**Astrophysics
**PHYS-UA
150

*Prerequisite: Physics III (PHYS-UA 95) or permission of the instructor. Offered every other year. 4 points.*

Introduction to modern astrophysical problems with an emphasis on the physical concepts involved: radio, optical, and X-ray astronomy; stellar structure and evolution; white dwarfs, pulsars, and black holes; and galaxies, quasars, and cosmology.

**Physics
of Biology
**PHYS-UA
160

*Prerequisite: Physics III (PHYS-UA 95). Offered every other year. 3 points.*

Basic biological processes at all levels of organization (molecular, cellular, organismal, and population) in the light of simple ideas from physics. Topics include self-assembly, molecular motors, low Reynolds fluid dynamics, optical imaging, and single-molecule manipulation. Intended for students with a background in mathematics and the physical sciences.

**General
Relativity
**PHYS-UA
170

*Prerequisite: Dynamics (PHYS-UA 120) or permission of the instructor. Offered every other year in the spring. 3 points.*

Provides an introduction to general relativity, stressing physical phenomena and their connection to experiments and observations. Topics include special relativity, gravity as geometry, black holes, gravitational waves, cosmology, Einstein equations.

**Introduction
to Fluid Dynamics
**PHYS-UA
180

*Identical to MATH-UA 230. Prerequisite: Calculus III (MATH-UA 123); Mathematical Physics (PHYS-UA 106) is recommended. Offered every year. 4 points.*

Key concepts of fluid dynamics: the formalism of continuum mechanics, the conservation of mass, energy and momentum in a fluid, the Euler and Navier-Stokes equations, and viscosity and vorticity. Concepts are applied to such classic problems as potential flow around a cylinder, the Stokes flow, the propagation of sound and gravity waves, and the onset of instability in shear flow.

**Philosophy of Physics**

PHYS-UA 190 *Identical
to ** PHIL-UA 94. *Offered every other year. 4 points.

We will investigate different approaches to understanding space and time, and how the account of space-time structure has evolved in physics. One of the main objectives is to have a clear and accurate understanding of the Special Theory of Relativity, detailed enough to allow the student to solve some physics problems. This will require a bit of mathematics, but not more than algebra. We will discuss the General Theory of Relativity in a more qualitative way, including an account of the structure of black holes. Philosophy students do not need any further background in physics or mathematics, and physics students will not benefit from greater mathematical sophistication. We will also study the relevant history of physics and philosophy, particularly the debate between Newton and Leibniz about the nature of space and time. There will be two lectures each week and a recitation section.

**Computational
Physics
**PHYS-UA
210

*Prerequisites: Mathematical Physics (PHYS-UA 106) or permission of the instructor, and knowledge of a scientific programming language (such as C, C++, Fortran, or Python). Offered every year. 4 points.*

Introduction to computational physics, with an emphasis on fields of current research interest in which numerical techniques provide unique physical insight. Topics are chosen from various branches of physics, including numerical solution of ordinary and partial differential equations, eigenvalue problems, Monte Carlo methods in statistical mechanics, field theory, dynamical systems, and chaos.

**Special
Topics in Physics
**PHYS-UA
800

*Prerequisites vary with the topic. Offered occasionally. 3 points.*

Covers advanced topics or recent developments in physics. Detailed course descriptions are made available when topics are announced.

**Independent
Study
**PHYS-UA
997, 998

*Prerequisite: permission of the director of undergraduate studies. Offered in the fall and spring respectively. 2 to 4 points per term.*