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Conservation laws and energy transformations in a class of common physics problems
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 87 (No. 11) |
| Halaman | : | 868-874 |
| Abstrak | : | We analyze a category of problems that is of interest in many physical situations, including those encountered in introductory physics classes: systems with two well-delineated parts that exchange energy, eventually reaching a shared equilibrium with a loss of mechanical or electrical energy. Such systems can be constrained by a constant of the system (e.g., mass, charge, momentum, or angular momentum) that uniquely determines the mechanical or electrical energy of the equilibrium state, regardless of the dissipation mechanism. A representative example would be a perfectly inelastic collision between two objects in one dimension, for which momentum conservation requires that some of the initial kinetic energy is dissipated by conversion to thermal or other forms as the two objects reach a common final velocity. We discuss how this feature manifests in a suite of four well-known and disparate problems that all share a common mathematical formalism. These examples, in which the energy dissipated during the process can be difficult to solve directly from dissipation rates, can be approached by students in a first-year physics class by considering conservation laws and can therefore be useful for teaching about energy transformations and conserved quantities. We then illustrate how to extend this method by applying it to a final example. |
Finite and half-infinite solenoids and the Aharonov-Bohm effect
| Pengarang | : | Suroso |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 87 (No. 11) |
| Halaman | : | 862-867 |
| Abstrak | : | In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, ?, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations. |
Quantum matrix diagonalization visualized
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 87 (No. 11) |
| Halaman | : | 857-861 |
| Abstrak | : | We show how to visualize the process of diagonalizing the Hamiltonian matrix to find the energy eigenvalues and eigenvectors of a generic one-dimensional quantum system. Starting in the familiar sine-wave basis of an embedding infinite square well, we display the Hamiltonian matrix graphically with the basis functions alongside. Each step in the diagonalization process consists of selecting a nonzero off-diagonal matrix element and then rotating the two corresponding basis vectors in their own subspace until this element is zero. We provide mathematica code to display the effects of these rotations on both the matrix and the basis functions. As an electronic supplement, we also provide a javascript web app to interactively carry out this process. |
Simulating gravitational motion, gas dynamics, and structure in the cosmos
| Pengarang | : | Skomra, Piotr A.,Keyser, John D. |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 478-485 |
| Abstrak | : | We provide introductory explanations and illustrations of the N-body hydrodynamics code Charm N-body GrAvity solver (ChaNGa). ChaNGa simulates the gravitational motion and gas dynamics of matter in space with the goal of modeling galactic and/or cosmological structure and evolution. We discuss the algorithm for leapfrog integration and smoothed particle hydrodynamics and computer science concepts used by the program, including the binary data structure for the particle positions. Our presentation borrows from the doctoral dissertation of J. Stadel, U. Washington, 2001. Problems are provided in order to use ChaNGa to learn or solidify some cosmological concepts. |
Cost-effective measurement of magnetostriction in nanoscale thin films through an optical cantilever displacement method
| Pengarang | : | David L. Tran |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 470-477 |
| Abstrak | : | A cost-effective method for the quantitative characterization of the magnetostrictive effect in thin films is presented. In this method, a sample's magnetostriction is extrapolated from the tip displacement of a thin-film magnetostrictive cantilever. The tip displacement is measured by monitoring the position of a reflected laser beam using two differentially coupled photodiode positioning sensors. In contrast with alternative optical deflection-angle devices designed for educational purposes, the detection limit of our setup resolves submicron-level displacements from nanoscale thin films. The efficacy of the system is demonstrated through measurements using amorphous 200-nm thick Terfenol-D/Si (100) bimorph cantilevers. In these measurements, magnetostriction values of  ppm at ±4300 Oe applied field were attained, where the voltage noise floor was ±0.05 V (a cantilever displacement uncertainty of ±70 nm). In-plane (IP) and out-of-plane (OOP) magnetization curves and crystallographic x-ray diffraction (XRD) were performed to determine the magnetic behavior and confirm the amorphous nature of the films, respectively. The experimental methods and material characterization systems demonstrated here enhance the understanding of complex magnetic phenomena and introduce common measurement techniques to better equip students with the skills for insightful analysis of fundamental magnetic physics. |
Free expansion of a Gaussian wavepacket using operator manipulations
| Pengarang | : | Hunter, Kara E.,Bloch, Rebecca I.,Kleinman, Gary |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 463-469 |
| Abstrak | : | The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing. |
Endless fun in high dimensions—A quantum card game
| Pengarang | : | Lea Kopf |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 458-462 |
| Abstrak | : | Quantum technologies enable new ways to distribute and process information. The enormous progress over the recent decades has led to an urgent need for new educational programs to train professionals to work in this field. Here, we present a card game that teaches students the building blocks of quantum computing through strategic gameplay. Participants start from the lowest quantum state and play cards that change their state and/or their opponents' state, aiming to build an algorithm that achieves the highest possible quantum state. Players can utilize several different strategies that rely on quantum features such as randomness, superposition, interference, and entanglement. Our game expands on the existing Q game, originally developed using traditional qubits (with 2-level states), by including an option to play with qutrits (with 3-level states), and by developing cooperative and single player modes in addition to the existing competitive mode. The presented game contributes to the ongoing efforts on gamifying quantum physics education with a particular focus on the counter-intuitive features that make quantum computing powerful. |
Beyond the ABCDs: A better matrix method for geometric optics by using homogeneous coordinates
| Pengarang | : | Theodore A. Corcovilos |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 449-457 |
| Abstrak | : | Geometric optics is often described as tracing the paths of non-diffracting rays through an optical system. In the paraxial limit, ray traces can be calculated using ray transfer matrices (colloquially, ABCD matrices), which are 2 × 2 matrices acting on the height and slope of the rays. A known limitation of ray transfer matrices is that they only work for optical elements that are centered and normal to the optical axis. In this article, we provide an improved 3 × 3 matrix method for calculating paraxial ray traces of optical systems that is applicable to how these systems are actually arranged on the optical table: lenses and mirrors in any orientation or position (e.g., in laboratory coordinates), with the optical path zig-zagging along the table. Using projective duality, we also show how to directly image points through an optical system using a point transfer matrix calculated from the system's ray transfer matrix. We demonstrate the usefulness of these methods with several examples and discuss future directions to expand the applications of this technique. |
A magnet falling inside a conducting pipe: Dependence of the drag force on the magnet orientation
| Pengarang | : | Yeter, Ibrahim H,Aguirre-Muñoz, Zenaida,Garro, Elias S. Loria,Koca, Fatih |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 440-448 |
| Abstrak | : | We develop a simple model to investigate the orientation-dependence of the drag force acting on a magnet falling inside a vertical conducting pipe. We approximate the magnet by a point magnet and the pipe by a two-dimensional cylindrical surface. Independent of the magnet's orientation, the drag force is proportional to its velocity: ?. We show that the coefficient of the horizontally oriented magnet is about 2/3 of the coefficient for the vertically oriented magnet. If the magnetic moment makes an angle θ with the vertical direction, the drag coefficient k can be expressed as ?. When the magnet falls with a non-vertical orientation, a local charge distribution is induced in the pipe, which plays a role as important as that of the time-varying magnetic field due to the falling magnet in generating the eddy currents. The model's predictions are compared with experimental results. |
Investigating the magnetic field outside small accelerator magnet analogs via experiment, simulation, and theory
| Pengarang | : | Kelley D. Sullivan |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 91 (No. 6) |
| Halaman | : | 432-439 |
| Abstrak | : | Particle accelerators use powerful and complex magnetic fields to turn, shape, and eventually collide beams of near-light-speed particles, yet the fundamental magnetic principles behind the accelerator magnets can be understood by undergraduate students. In this paper, we use small-scale accelerator magnet analogs in a multi-faceted, low-cost exploration of the magnetic field exterior to accelerator magnets. These fields are best understood using the multipole expansion of the field. If we assume that the magnetic field is created by ideal magnetic dipoles, we can derive a theoretical model that shows that each accelerator magnet configuration is dominated by a single multipole moment and obeys ?, where l is the multipole order (with for the dipole, quadrupole, octopole, and hexadecapole moments, respectively). Using commercially available NdFeB magnets and the magnetic field sensor inside a smartphone, we experimentally verify the power-law dependence of the accelerator magnet configurations. Finally, we use the open-source Python library Magpylib to simulate the magnetic field of the permanent magnet configurations, showing good agreement among theory, experiment, and simulation. |