Volume 81, Issue 11, November 2013

A cone of light appears in a tank of water when a laser pointer shines through the water onto a white piece of paper upon which the tank is sitting. We describe how students can understand the origins of this cone by constructing multiple explanations, then proposing and designing experiments to test their explanations. This process is the foundation of the Investigative Science Learning Environment (ISLE) framework, designed to engage students in the reasoning activities similar to those that physicists use to construct and apply new knowledge. We describe typical student ideas and provide a list of equipment and suggestions for facilitating student exploration relating to optics. We also explain the formal physics behind the phenomena that are involved in the experiment. Finally, we suggest how the ISLE framework can be used to help instructors find problems and experiments that engage students in devising and testing multiple explanations.
 PAPERS


Resource Letter MPCVW1: Modeling Political Conflict, Violence, and Wars: A Survey
View Description Hide DescriptionThis Resource Letter provides a guide into the literature on modeling and explaining political conflict, violence, and wars. Although this literature is dominated by social scientists, multidisciplinary work is currently being developed in the wake of myriad methodological approaches that have sought to analyze and predict political violence. The works covered herein present an overview of this abundance of methodological approaches. Since there is a variety of possible data sets and theoretical approaches, the level of detail and scope of models can vary quite considerably. The review does not provide a summary of the available data sets, but instead highlights recent works on quantitative or multimethod approaches to modeling different forms of political violence. Journal articles and books are organized in the following topics: social movements, diffusion of social movements, political violence, insurgencies and terrorism, and civil wars.

A simple optics experiment to engage students in scientific inquiry
View Description Hide DescriptionA cone of light appears in a tank of water when a laser pointer shines through the water onto a white piece of paper upon which the tank is sitting. We describe how students can understand the origins of this cone by constructing multiple explanations, then proposing and designing experiments to test their explanations. This process is the foundation of the Investigative Science Learning Environment (ISLE) framework, designed to engage students in the reasoning activities similar to those that physicists use to construct and apply new knowledge. We describe typical student ideas and provide a list of equipment and suggestions for facilitating student exploration relating to optics. We also explain the formal physics behind the phenomena that are involved in the experiment. Finally, we suggest how the ISLE framework can be used to help instructors find problems and experiments that engage students in devising and testing multiple explanations.

A random walk to stochastic diffusion through spreadsheet analysis
View Description Hide DescriptionThis paper describes a random walk simulation using a number cube and a lattice of concentric rings of tiled hexagons. At the basic level, it gives beginning students a concrete connection to the concept of stochastic diffusion and related physical quantities. A simple algorithm is presented that can be used to set up spreadsheet files to calculate these simulated quantities and even to “discover” the diffusion equation. Lattices with different geometries in two and three dimensions are also presented. This type of simulation provides fertile ground for independent investigations by all levels of undergraduate students.

Mutual inductance between piecewiselinear loops
View Description Hide DescriptionWe consider a currentcarrying wire loop made out of linear segments of arbitrary sizes and directions in threedimensional space. We develop expressions to calculate its vector potential and magnetic field at all points in space. We then calculate the mutual inductance between two such (nonintersecting) piecewiselinear loops. As simple applications, we consider in detail the mutual inductance between two square wires of equal length that either lie in the same plane or lie in parallel horizontal planes with their centers on the same vertical axis. Our expressions can also be used to obtain approximations to the mutual inductance between wires of arbitrary threedimensional shapes.

How does a magnetic trap work?
View Description Hide DescriptionMagnetic trapping is a cornerstone of modern ultracold physics and its applications, including quantum information processing, quantum metrology, quantum optics, and highresolution spectroscopy. Here, a comprehensive analysis and discussion of the basic physics behind the most commonly used magnetic traps used in BoseEinstein condensation is presented. This analysis includes the quadrupole trap, the timeaveraged orbiting potential trap, and the IoffePritchard trap. The trapping conditions and efficiency of these devices can be determined from simple derivations based on classical electromagnetism, even though they operate on quantum objects.

On the connection between image formation formulas in geometrical optics and beam transformation formulas in wave optics
View Description Hide DescriptionThe close connection between image formation in geometrical optics and beam transformation by a paraxial optical system is examined analytically using mathematical tools accessible to undergraduate students, such as the Fresnel diffraction integral and Fourier transforms, instead of the more complicated Wigner distribution or coherence functions frequently employed in the literature. It is shown that geometrical optics correctly predicts the plane where a beam is refocused and its magnification only for afocal optical systems or in the limit of point sources. We illustrate this theory by simulating the transformation of a flattop beam by a pair of lenses.

A simple proof of Bell's inequality
View Description Hide DescriptionBell’s theorem is a fundamental result in quantum mechanics: it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of preexisting local properties. We give an extremely simple proof of Bell's inequality; a single figure suffices. This simplicity may be useful in the unending debate over what exactly the Bell inequality means, because the hypotheses underlying the proof become transparent. It is also a useful didactic tool, as the Bell inequality can be explained in a single intuitive lecture.

Exact solution of the compressed hydrogen atom
View Description Hide DescriptionThe exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,…), L (= 0, 1, 2,…), and M (= − L, − L + 1,…, L − 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the largesphere limit, recovering the typical n ^{2} degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong and weakconfinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed.
