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Exact solutions for the inverse problem of the time-independent Schrödinger equation
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| Pengarang | : | |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 975-979 |
| Abstrak | : | We introduce an “inverse method” for solving the time-independent Schrödinger equation. Rather than derive wave functions that are solutions for a given external potential ?(?→)?, we ask the inverse question of which ?(?→) will have a given probability density function ?(?→)?. Several examples of ground states in one, two, and three dimensions are presented for both well-known and more exotic probability density functions in position space. |
Hyperbolic times in Minkowski space
| Pengarang | : | An?l Zenginoglu |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 965-974 |
| Abstrak | : | Time functions with asymptotically hyperbolic geometry play an increasingly important role in many areas of relativity, from computing black hole perturbations to analyzing wave equations. Despite their significance, many of their properties remain underexplored. In this expository article, I discuss hyperbolic time functions by considering the hyperbola as the relativistic analog of a circle in two-dimensional Minkowski space and argue that suitably defined hyperboloidal coordinates are as natural in Lorentzian manifolds as spherical coordinates are in Riemannian manifolds. |
Counterintuitive properties of relativistic relative motion for accelerated observers
| Pengarang | : | M. Pössel |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 957-964 |
| Abstrak | : | A challenge in teaching special relativity is that a number of the theory's effects are at odds with the intuition of classical physics as well as students' everyday experience. The relativity of simultaneity, time dilation, and length contraction are prominent examples. This article describes two other counterintuitive properties of relative motion with accelerated observers: (a) when two objects have a constant radar distance and are by that standard “at rest,” their relative speed is not necessarily zero, and (b) for two observers A and B, it is possible for A to observe the two to be approaching each other, while B considers them to be moving away from each other. The two effects provide insight into static gravitational fields and horizons in general relativity. |
Completeness of energy eigenfunctions for the reflectionless potential in quantum mechanics
| Pengarang | : | Fatih Erman; O. Teoman Turgut |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 950-956 |
| Abstrak | : | There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians. |
Why is there no Poisson spot in a solar eclipse
| Pengarang | : | Jami J. Kinnunen |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 945-949 |
| Abstrak | : | The Poisson spot is a fascinating lecture demonstration. Its simple explanation can lead to further questions, not only the one posed in the title, but also questions such as why the simple model that considers only light passing just outside the spherical object is successful. The Huygens–Fresnel diffraction model is applied to answer these questions. |
Teaching magnetism with bivectors
| Pengarang | : | Steuard Jensen |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 936-944 |
| Abstrak | : | The magnetic field is traditionally presented as a (pseudo)vector quantity, tied closely to the cross product. Though familiar to experts, many students find these ideas challenging and full of subtleties. Building on earlier work in rotational physics, we present an alternative pedagogical approach that describes magnetic fields using bivectors. These objects can be visualized as oriented tiles whose components form an antisymmetric matrix. Historically, bivectors have been mostly used in specialized contexts like spacetime classification or geometric algebra, but they are not necessarily more complicated to understand than cross products. Teaching magnetism in this language addresses common student difficulties, generalizes directly to relativity (and extra dimensions), and brings fresh insight to familiar ideas. |
Energy density of any capacitor or inductor
| Pengarang | : | Jinliang Zheng; Shang-Yuu Tsai |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 931-935 |
| Abstrak | : | Most introductory physics courses derive the energy densities of the static electric and magnetic field for the simple cases of parallel plate capacitors and infinitely long solenoids. Authors then inform readers, usually without proof, that the energy density equations derived for the simple cases are actually the correct equations for all capacitors and inductors, regardless of their shape. The proof of the general case is typically omitted because it involves the differential calculus of vector fields. In view of this, we provide a derivation for the energy density only based on integral calculus for capacitors and inductors of any kind. The derivation, albeit seemingly complicated at first, is conceptually simple enough for introductory physics courses and does not require any knowledge of the differential calculus of vector fields. |
Scattering shadows
| Pengarang | : | Thomas Curtright; Gaurav Verma |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 12) |
| Halaman | : | 924-930 |
| Abstrak | : | We discuss the regions forbidden to classical scattering trajectories by repulsive potentials. We give explicit results for the asymptotic form of these regions, far from the scattering center, in terms of the scattering angle function. We account for divergent total cross sections on the basis of these results. |
An instructional lab apparatus for quantum experiments with single nitrogen-vacancy centers in diamond
| Pengarang | : | Zhiyang Yuan; Sounak Mukherjee; Jeff D. Thompson; Nathalie P. de Leon; Aedan Gardill; Shimon Kolkowitz |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 11) |
| Halaman | : | 892-900 |
| Abstrak | : | Hands-on experimental experience with quantum systems in the undergraduate physics curriculum provides students with a deeper understanding of quantum physics and equips them for the fast-growing quantum science industry. Here, we present an experimental apparatus for performing quantum experiments with single nitrogen-vacancy (NV) centers in diamond. This apparatus is capable of basic experiments such as single-qubit initialization, rotation, and measurement, as well as more advanced experiments investigating electron–nuclear spin interactions. We describe the basic physics of the NV center and give examples of potential experiments that can be performed with this apparatus. We also discuss the options and inherent trade-offs associated with the choice of diamond samples and hardware. The apparatus described here enables students to write their own experimental control and data analysis software from scratch, all within a single semester of a typical lab course, as well as to inspect the optical components and inner workings of the apparatus. We hope that this work can serve as a standalone resource for any institution that would like to integrate a quantum instructional lab into its undergraduate physics and engineering curriculum. |
Teaching labs for blind students: Equipment to measure the inertia of simple objects
| Pengarang | : | Alfredo Lisboa; Francisco J. Pena |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 92 (No. 11) |
| Halaman | : | 889-891 |
| Abstrak | : | This article describes a laboratory experiment for blind students to measure the moment of inertia of simple objects; in this case, that of a disc about its axis of symmetry. We adapted our usual lab to modify the data collection process, using an open-source electronic platform to convert visual signals into acoustic signals. This allowed blind students at our University to participate alongside their classmates in the mechanics lab. |