Physics

Physics—the study of matter and energy, time and space, and their interactions and interconnections—is often regarded as the most fundamental of the natural sciences. An understanding of physics is essential for an understanding of many aspects of chemistry, which in turn provides a foundation for understanding a variety of biological processes. Physics also plays an important role in most branches of engineering; and the field of astronomy, essentially, is physics applied on the largest of scales.

As science has progressed over the last century or so, the boundaries between the different scientific disciplines have become blurred, and new interdisciplinary fields—such as chemical physics, biophysics, and engineering physics—have arisen. For these reasons, and because of the excellent training in critical thinking and problem-solving provided by the study of physics, this subject represents an indispensable gateway to the other natural sciences and a valuable component of a liberal-arts education.

Physics 2021-2022 Courses

It’s About Time

Open, Small Lecture—Fall | 5 credits

This seminar will explore the topic of time from a wide variety of viewpoints—from the physical to the metaphysical to the practical. We will seek the answers to questions such as: What is time? How do we perceive time? Why does time appear to flow only in one direction? Is time travel possible? How is time relative? We will explore the perception of time across cultures and eras, construct an appreciation of the arrow of time by designing and building a Rube Goldberg machine, and discuss scientific articles and science-inspired works of fiction to make sense of this fascinating topic. Time stops for no one, but let’s take some time to appreciate its uniqueness.

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Classical Mechanics (Calculus-Based General Physics)

Open, Seminar—Fall | 5 credits

Calculus-based general physics is a standard course at most institutions; as such, this course will prepare you for more advanced work in the physical science, engineering, or health fields. This course will cover introductory classical mechanics, including kinematics, dynamics, momentum, energy, and gravity. Emphasis will be placed on scientific skills, including: problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. The best way to develop scientific skills is to practice the scientific process. We will focus on learning physics through discovering, testing, analyzing, and applying fundamental physics concepts in an interactive classroom, as well as in weekly laboratory meetings.

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Introduction to Mechanics (General Physics Without Calculus)

Open, Seminar—Fall | 5 credits

This course covers introductory classical mechanics, including dynamics, kinematics, momentum, energy, and gravity. Students considering careers in architecture or the health sciences, as well as those interested in physics for physics’ sake, should take either this course or Classical Mechanics. Emphasis will be placed on scientific skills, including problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. Seminars will incorporate discussion, exploratory activities, and problem-solving activities. In addition, the class will meet weekly to conduct laboratory work. A background in calculus is not required.

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Classical and Quantum Waves

Intermediate, Seminar—Fall | 5 credits

This course, which will provide an introduction to both classical and quantum waves, is a required prerequisite course for those interested in pursuing the Columbia Combined Plan program in applied mathematics, applied physics, biomedical engineering, electrical engineering, and materials science and engineering. Topics will include: classical waves and the wave equation, oscillations and normal modes, Fourier series and Fourier transforms; quantum waves and the Schrödinger equation; topics from quantum physics, including quantization of energy levels and reflection and transmission off barriers; and various applications of waves corresponding to student interests.

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Electromagnetism & Light (Calculus-Based General Physics)

Open, Seminar—Spring | 5 credits

Calculus-based general physics is a standard course at most institutions; as such, this course will prepare you for more advanced work in the physical science, engineering, or health fields. This course will cover waves, geometric and wave optics, electrostatics, magnetostatics, and electrodynamics. We will use the exploration of the particle and wave properties of light to bookend our discussions and ultimately finish our exploration of classical physics with the hints of its incompleteness. Emphasis will be placed on scientific skills, including: problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. The best way to develop scientific skills is to practice the scientific process. We will focus on learning physics through discovering, testing, analyzing, and applying fundamental physics concepts in an interactive classroom, as well as in weekly laboratory meetings.

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20th-Century Physics

Open, Seminar—Spring | 5 credits

This course will provide an overview of the pivotal developments in 20th-century physics that dramatically overturned the centuries-old scientific understanding of the fundamental laws of our universe. In this seminar-style class, we will discuss readings, walk through thought experiments, and unravel paradoxes to understand the concepts behind Einstein’s theories of special and general relativity, debate various interpretations of quantum mechanics, and explore the open questions that are motivating theoretical physics research in the 21st century.

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Introduction to Electromagnetism, Light, and Modern Physics (General Physics Without Calculus)

Intermediate, Seminar—Spring | 5 credits

This course covers electromagnetism and optics. Emphasis will be placed on scientific skills, including problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. Seminars will incorporate discussion, exploratory, and problem-solving activities. In addition, the class will meet weekly to conduct laboratory work.

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First-Year Studies: Chemistry for Contrarians: A Nontraditional Science Course for Liberal Arts Students

Open, FYS—Year

For anyone who wants to know how the world (and the universe) works at a fundamental level, modern science has (almost) all the answers; however, painful memories of school science classes and seemingly impenetrable scientific jargon often put people off from engaging with this area of study. In this course, we will take two very different approaches to engage with chemistry and related areas of physics and biology. I hope to convince you that science is, ultimately, about people—how we learn about and change our beliefs concerning the physical world. Fall semester: Gaming Our Way to Scientific Literacy. In recent years, a number of educational board and card games have been designed to aid students in learning the vocabulary and concepts of the physical and life sciences. The manufacturers of these games claim that they are scientifically accurate and offer a novel way for nontraditional learners to develop a working knowledge of basic science. We will study a number of important core topics in subatomic and atomic physics, chemistry, and biochemistry. To enliven our classes, we will use the following games as the center of each unit of study: Subatomic: An Atom Building Game™; Periodic: A Game of the Elements™; Covalence: A Molecule Building Game™; Ion: A Compound Building Game™; Peptide: A Protein Building Game™; and Cytosis: A Cell Biology Game™. In each case, we will look at how the developers have integrated current scientific knowledge into their games. By playing, we will determine how effective these games are in helping us to learn scientific concepts and to gain confidence using scientific vocabulary. Spring Semester: Reading and Writing the Biography of Chemistry. During the spring semester, we will read the stories of some chemical elements and important chemical compounds—not just their discovery but also their cultural and historical significance. We will discover how different cultures affect attitudes toward various chemicals and their use and how, in return, important chemicals have affected culture and transformed lives. During the fall semester, students will meet with the instructor weekly for individual conferences. In the spring, we will meet weekly or every other week, depending on students’ needs and the progress of their conference projects.

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General Chemistry I

Open, Small Lecture—Fall

Chemistry is the study of the properties, composition, and transformation of matter. Chemistry is central to the production of the materials required for modern life; for example, the synthesis of pharmaceuticals to treat disease, the manufacture of fertilizers and pesticides required to feed an ever-growing population, and the development of efficient and environmentally benign energy sources. This course provides an introduction to the fundamental concepts of modern chemistry. We will begin by examining the structure and properties of atoms, which are the building blocks of the elements and the simplest substances in the material world around us. We will then explore how atoms of different elements can bond with each other to form an infinite variety of more complex substances, called compounds. This will lead us to an investigation of several classes of chemical reactions: the processes by which substances are transformed into new materials with different physical properties. Along the way, we will learn how and why the three states of matter (solids, liquids, and gases) differ from one another and how energy may be either produced or consumed by chemical reactions. In weekly laboratory sessions, we will perform experiments to illustrate and test the theories presented in the lecture part of the course. These experiments will also serve to develop practical skills in both synthetic and analytic chemical techniques.

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General Chemistry II

Intermediate, Small Lecture—Spring

This course is a continuation of General Chemistry I. We will begin with a detailed study of both the physical and chemical properties of solutions, which will enable us to consider the factors that affect both the rates and direction of chemical reactions. We will then investigate the properties of acids and bases and the role that electricity plays in chemistry. The course will conclude with introductions to nuclear chemistry and organic chemistry. Weekly laboratory sessions will allow us to demonstrate and test the theories described in the lecture segment of the course.

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Green Infrastructure

Intermediate, Seminar—Fall

Green infrastructure has the potential to transform our cities, replacing asphalt and concrete with soil, vegetation, and waterways. But while cities across the globe are now developing green infrastructure plans to protect water resources, enhance biodiversity, and adapt to the impacts of global climate change, there is an ongoing debate on what green infrastructure actually is. And there are still many remaining barriers to its broad implementation in our cities and suburbs. In this seminar, we will explore green infrastructure through the lens of ecosystem services—the regulating, provisioning, and cultural benefits that natural ecosystems provide for free to humans. Through quantitative case studies and field visits to green infrastructure projects in Yonkers and New York City, we will learn about a variety of different types of green infrastructure, including rain gardens, green roofs, detention basins, and constructed wetlands. We will also learn about the challenges associated with assessing the performance of green infrastructure and will critically evaluate existing green infrastructure plans and designs.

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An Introduction to Statistical Methods and Analysis

Open, Lecture—Spring

Variance, correlation coefficient, regression analysis, statistical significance, and margin of error...you’ve heard these terms and other statistical phrases bantered about before, and you’ve seen them interspersed in news reports and research articles. But what do they mean? And why are they so important? Serving as an introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental methods of statistical analysis used to gain insight into diverse areas of human interest. The use, misuse, and abuse of statistics will be the central focus of the course; and specific topics of exploration will be drawn from experimental design theory, sampling theory, data analysis, and statistical inference. Applications will be considered in current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical (spreadsheet) software will be introduced and used extensively in this course, but no prior experience with the technology is assumed. Group conferences, conducted in workshop mode, will serve to reinforce student understanding of the course material. This lecture is recommended for anybody wishing to be a better-informed consumer of data and strongly recommended for those planning to pursue advanced undergraduate or graduate research in the natural sciences or social sciences.

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Calculus I: The Study of Motion and Change

Open, Seminar—Fall and Spring

Our existence lies in a perpetual state of change. An apple falls from a tree; clouds move across expansive farmland, blocking out the sun for days; meanwhile, satellites zip around the Earth, transmitting and receiving signals to our cell phones. The calculus was invented to develop a language to accurately describe and study the change that we see. The ancient Greeks began a detailed study of change but were scared to wrestle with the infinite; so it was not until the 17th century that Isaac Newton and Gottfried Leibniz, among others, tamed the infinite and gave birth to this extremely successful branch of mathematics. Though just a few hundred years old, the calculus has become an indispensable research tool in both the natural and social sciences. Our study begins with the central concept of the limit and proceeds to explore the dual topics of differentiation and integration. Numerous applications of the theory will be examined. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind.

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Calculus II: Further Study of Motion and Change

Open, Seminar—Spring

This course continues the thread of mathematical inquiry, following an initial study of the dual topics of differentiation and integration (see Calculus I course description). Topics to be explored in this course include the calculus of exponential and logarithmic functions, applications of integration theory to geometry, alternative coordinate systems, infinite series, and power series representations of functions. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study of some other mathematically-related topic. This seminar is intended for students interested in advanced study in mathematics or science, for those preparing for careers in the health sciences or engineering, or for any simply wishing to broaden and enrich the life of the mind.

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Multivariable Mathematics: Linear Algebra, Vector Calculus, and Differential Equations

Intermediate, Seminar—Year

Rarely is a quantity of interest—tomorrow’s temperature, unemployment rates across Europe, the cost of a spring-break flight to Fort Lauderdale—a simple function of just one primary variable. Reality, for better or worse, is mathematically multivariable. This course introduces an array of topics and tools used in the mathematical analysis of multivariable functions. The intertwined theories of vectors, matrices, and differential equations and their applications will be the central themes of exploration in this yearlong course. Specific topics to be covered include the algebra and geometry of vectors in two, three, and higher dimensions; dot and cross products and their applications; equations of lines and planes in higher dimensions; solutions to systems of linear equations, using Gaussian elimination, theory and applications of determinants, inverses and eigenvectors, volumes of three-dimensional solids via integration, spherical and cylindrical coordinate systems, and methods of visualizing and constructing solutions to differential equations of various types. Conference work will involve an investigation of some mathematically-themed subject of the student’s choosing.

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Mathematics in Theory and Practice: Real Analysis and Topology

Intermediate, Seminar—Year

The calculus of Newton and Leibniz was so successful that science forgave the logical shortcomings of its “fluxions” and “evanescent quantities.” In the 19th century, however, calculus evolved into the study of functions of a real variable—real analysis—which is a model of the foundational rigor that has come to define mathematics as a discipline. In the 20th century, the search for axioms of the real numbers uncovered subtle assumptions about spatial properties of the real line. These properties—such as continuity, separability, and dimension—do not depend on magnitude but on more general notions of position. The geometry of position, or topology as it is called today, is the study of exactly such properties. This yearlong seminar will begin with preliminaries of discrete mathematics, including symbolic logic, proof technique, and set theory. We will study these topics in the context of networks and surfaces, which are some of the most intuitive topological objects. This will be followed by an in-depth study of the real numbers, sequences and series, limits, continuity, the derivative, and the integral. To motivate our revision of these familiar calculus terms, the seminar will read and discuss important counterexamples, such as nowhere-differentiable continuous functions, rearrangements of infinite series, and the Cantor set. At the end of the year, we will return to topology. This will give us the opportunity to see how many of the geometric properties of curves, surfaces, and maps between them find a unified expression in terms of relations among point sets. Conference work will clarify seminar ideas and possibly their application to mathematical models in the natural sciences, computer science, or economics.

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First-Year Studies: The Senses: Art and Science

Open, FYS—Year

The perceiving mind is an incarnated mind. —Maurice Merleau-Ponty, 1964

Sensory perception is a vital component of the creation and experience of artistic works of all types. Investigation of sensory systems has been foundational for psychologists and neuroscientists developing understanding of brains, minds, and bodies. Recent work in brain science has moved us beyond the Aristotelian notion of five discrete senses to a view of the senses as more various and interconnected—with each other and with the fundamental psychological categories of perception, attention, emotion, memory, imagination, and judgment. What we call “taste” is a multisensory construction of “flavor” that relies heavily on smell, vision, and touch (mouth feel); “vision” refers to a set of semi-independent streams that specialize in the processing of color, object identity, or spatial layout and movement; “touch” encompasses a complex system of responses to different types of contact with the largest sensory organ—the skin; and “hearing” includes aspects of perception that are thought to be quintessentially human—music and language. Many other sensations are not covered by the standard five: for example, the senses of balance, of body position (proprioception) and ownership, feelings of pain arising from within the body, and feelings of heat or cold. Perceptual psychologists have suggested that the total count is closer to 17 than five. We will investigate all of these senses, their interactions with each other, and their intimate relationships with human emotion, memory, and imagination. Some of the questions that we will address are: Why are smells such potent memory triggers? What can visual art tell us about how the brain works and vice versa? Why is a caregiver’s touch so vital for psychological development? Why do foods that taste sublime to some people evoke feelings of disgust in others? Do humans have a poor sense of smell (and have the effects of COVID-19 changed our views of its importance)? Why does the word “feeling” refer to both bodily sensations and emotions? What makes a song “catchy” or “sticky”? Can humans learn to echolocate like bats? What is the role of body perception in mindfulness meditation? This is a good course for artists who like to think about science and for scientists with a feeling for art. This is a collaborative course, with small-group meetings held weekly in addition to the individual conference meetings held every other week. The main small-group, collaborative activity is a sensory lab where students will have the opportunity to explore their own sensory perceptions in a systematic way, investigating how they relate to language, memory, and emotion. Other group activities include mindful movement and other meditation practices for stress relief and emotional regulation, as well as occasional museum visits if these can be done safely.

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Theories of Agency and Action in Science Studies

Open, Lecture—Fall

This course surveys a rich historical debate in science, technology, and society studies on the nature of agency—or the motivation behind, and responsibility for, action. The lecture course begins with an exploration of the nature of scientific fact, including how discoveries are made and how they become accepted in society. We will pay special attention to the concepts of co-production, the idea that humans and technologies work together, and situated action, the reality that actions are rooted in social context, to study how technologies become central to social interaction. This grounding theory will lay a foundation for students to consider an ongoing debate on the distinction between human and nonhuman action. The course culminates with an exploration of three contemporary discussions on the nature of agency with respect to automated weapons systems, assistive technologies for people with disabilities, and the use of algorithms to order social life. For each topic, we will consider how technologies influence social interaction and who or what is responsible when things go wrong. In group conference, students will practice analyzing how technologies shape social interaction through a series of “object readings,” short analyses of a single technological object. These assignments are designed to prepare students for a final group analysis of a technology of their choice.

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