Papers by Todd Timberlake
This book began as a college course. Astronomy 120: The Copernican Revolution was originally deve... more This book began as a college course. Astronomy 120: The Copernican Revolution was originally developed by my colleague Paul Wallace. The course was intended to fulfill a science requirement for students majoring in non-scientific fields at Berry College in Georgia (USA), where Paul and I taught. Paul's course was a great success, particularly when he taught it as a summer international course with stops in Poland, the Czech Republic, and Italy. Then, rather suddenly, Paul decided to leave Berry. The decision was shocking if you didn't know Paul. It's not typical for a physicist to give up a tenured faculty position to go to seminary, but that's what Paul did and, for him, it made sense. His decision, though, left me in some difficulty. First and foremost my friend and mentor was leaving, but his departure caused some practical problems too. Paul had been the only person to teach astronomy at Berry for many years. Should we hire an astronomer to replace Paul? It made more sense for the department to hire a physicist, but we needed someone to teach astronomy. I made what was, in hindsight, a rash decision, but one that I have never regretted. I decided that I would teach the astronomy courses at Berry. Although my doctoral degree is in physics, my undergraduate degree was in both physics and astronomy. I was confident that I could do it, but I only wanted to do it if I got to teach The Copernican Revolution. I had fallen in love with Paul's course. I loved it because it told a story, a story of science. Human beings love stories, but most students are never exposed to the story of science. They are taught science from a dry textbook that reads more like an encyclopedia than a novel. Teaching the history of science brings science to life. It shows students that science is a human activity, driven by the human passion to understand. It makes science interesting not just for the scientific knowledge, but for the struggle to gain that knowledge. Moreover, I thought Paul's course was exactly what nonscience majors needed. Students who plan to be scientists need training in the most current xi
Physics and Computation
Undergraduate lecture notes in physics, 2016
By this point we have seen that Maxima’s built-in routines can be very helpful tools in solving p... more By this point we have seen that Maxima’s built-in routines can be very helpful tools in solving physics problems. Users can expand Maxima’s usefulness by taking advantage of its programming capabilities. Although Maxima should not be viewed as a substitute for a full-featured programming language, it does have some basic programming features that allow users to write simple programs. This chapter introduces these basic programming features and shows how they can be used to solve problems in mathematics and physics.
Nonlinearity and Chaos
Undergraduate lecture notes in physics, 2016
One of the most exciting areas of current research in classical mechanics is the dynamic behavior... more One of the most exciting areas of current research in classical mechanics is the dynamic behavior of nonlinear systems. Nonlinear systems exhibit a much richer variety of behaviors than do linear systems. However, most nonlinear systems are impossible to solve using paper and pencil methods. It was only with the advent of digital computers that nonlinear dynamics really came into its own. With computer software like Maxima we can explore nonlinear dynamics in a way that is impossible without a computer.
Newtonian Mechanics
Undergraduate lecture notes in physics, 2016
arXiv (Cornell University), Dec 15, 2011
In 1785 astronomer William Herschel mapped out the shape of the Milky Way star system using measu... more In 1785 astronomer William Herschel mapped out the shape of the Milky Way star system using measurements he called "star-gages." Herschel's star-gage method is described in detail, with particular attention given to the assumptions on which the method is based. A computer simulation that allows the user to apply the star-gage method to several virtual star systems is described. The simulation can also be used to explore what happens when Herschel's assumptions are violated. This investigation provides a modern interpretation for Herschel's map of the Milky Way and why it failed to accurately represent the size and shape of our galaxy.
Oscillations
Undergraduate lecture notes in physics, 2016
Basic Newtonian Physics with Maxima
Undergraduate lecture notes in physics, 2016
Classical mechanics is the branch of physics that deals with the motion of objects subject to for... more Classical mechanics is the branch of physics that deals with the motion of objects subject to forces and constraints. Classical mechanics has existed as a well-defined subject since the publication of Newton’s Principia Mathematica Philosophiae Naturalis in 1687. Since Newton, a few important new concepts have been introduced into the subject (such as energy and its conservation), but most of the developments in classical mechanics since the seventeenth century have consisted of new mathematical techniques for solving classical mechanics problems.
Erratum: Nonlinearity and Chaos
Undergraduate lecture notes in physics, 2016
Undergraduate lecture notes in physics, 2016
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The one-dimensional potential energy function that is analogous to a two-dimensional track
American Journal of Physics, May 1, 2023
When learning about potential energy functions, students are sometimes told that the potential en... more When learning about potential energy functions, students are sometimes told that the potential energy function is analogous to that of a particle sliding along a frictionless roller coaster track or wire confined to a vertical plane with peaks and valleys of the track corresponding to unstable and stable equilibrium points. However, motion along a track with height z(x) is a constrained two-dimensional motion, not a one-dimensional motion, so the exact nature of this analogy may be unclear. We show that the horizontal motion of a point mass sliding along a frictionless track z(x) and subject to a uniform gravitational field is equivalent to the motion of a particle in one dimension characterized by an “analogous potential energy” function UE(x), which generally depends on the total energy of the system (and thus on the initial conditions). We derive a general expression for UE(x) in terms of z(x) and the total energy and show that the equilibrium points of the actual potential energy U(x)=mgz(x) are also static equilibrium points for UE(x) with the same stability. However, UE(x) may have additional dynamic equilibrium points that are not present for U(x). As an example, we derive UE(x) for a double well track and determine the period of oscillations on that track. The results show that in general a single track corresponds to many different analogous potential energy functions, each with a different value for the total energy.
An Earth-centered cosmos: astronomy and cosmology from
Martin Beech, Going Underground: The Science and History of Falling through the Earth. New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai and Tokyo: World Scientific, 2019. Pp. xi + 276. ISBN 978-9813-2790-3-2. £35.00/$38.00 (paperback)
The British Journal for the History of Science, 2020
elder, Dickens was a liberal Anglican, whereas Eliot translated German higher criticism and Geoff... more elder, Dickens was a liberal Anglican, whereas Eliot translated German higher criticism and Geoffroy was a deist. It would be well worth a future project to explore these connections between a priori world views and approaches to the natural history of humanity, but that would make for another book. In conclusion, Duncan’s work is readable and well researched, with Chapters 1 to 3 being particularly strong in demonstrating the interplay between natural philosophy, fiction, facts and imagination. H.-F. DESSAIN Independent scholar
Physical Review E
Statistical properties of the sequence of energy eigenvalues in a quantum system are related to t... more Statistical properties of the sequence of energy eigenvalues in a quantum system are related to the classical dynamics of the system. Sequences of energy levels for classically regular systems generally exhibit Poisson statistics, while those for classically chaotic systems exhibit random matrix statistics. For many systems the classical dynamics can change from regular to chaotic as the result of changing a system parameter and the quantum system shows a corresponding change in its eigenvalue statistics. Here we present a simple quantum system, composed of an infinite square well with several Dirac delta barriers placed inside it, that exhibits a change in eigenvalue statistics as the transmission probability T through the delta barriers changes. For T ≈0 the distribution of level spacings is Poisson-like, but increasing T causes the distribution to change to match that of the Gaussian Orthogonal Ensemble. For even greater T values the level spacing distribution becomes Gaussian with a standard deviation that approaches zero as T →1. A similar transition in statistics is seen in the number variance for the eigenvalue sequence. We also demonstrate that the structure of the energy eigenstates changes from highly localized at T ≈ 0 to evenly spread across the infinite square well as T → 1. Our model system can be viewed as an example of a quantum graph, and we show that our results match with the expectations for quantum graphs in the limit T → 0.
American Journal of Physics
Periodic quantum systems often exhibit energy spectra with well-defined energy bands separated by... more Periodic quantum systems often exhibit energy spectra with well-defined energy bands separated by band gaps. The formation of band structure in periodic quantum systems is usually presented in the context of Bloch's theorem or through other specialized techniques. Here we present a simple model of a finite one-dimensional periodic quantum system that can be used to explore the formation of band structure in a straightforward way. Our model consists of an infinite square well containing several evenly-spaced identical Dirac delta wells. Both attractive and repulsive delta wells are considered. We solve for the energy eigenvalues and eigenfunctions of this system directly and show the formation of band structure as the number of delta wells is increased, as well as how the size of the bands and gaps depends on the strength of the delta wells. These results are compared ot the predictions from Bloch's theorem. In addition, we use this model to investigate how the energy spectrum is altered by the introduction of two types of defects in the periodicity of the system. Strength defects, in which the strength of one delta well is changed, can result in an energy level moving from one band, through the band gap, to another band as the strength of the well is varied. Position defects, in which the location of one delta well is changed, can modulate the size of the energy bands and sufficiently large position defects can move an energy level into a gap. Band structure and defects are important concepts for understanding many properties of quantum solids and this simple model provides an elementary introduction to these ideas.
Two-dimensional Surfaces as Analogs for Potential Energy Functions
Bulletin of the American Physical Society, Mar 15, 2021
Moving the Earth: the revolutions of Copernicus
Finding our Place in the Solar System, 2019
Two spheres: modeling the heavens and the Earth
Finding our Place in the Solar System, 2019
How External Boundaries Affect the Number of Bound States in a Quantum Well
Bulletin of the American Physical Society, Mar 15, 2021
The Forest of Physics
American Journal of Physics, 2022
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Papers by Todd Timberlake