American Journal of Physics Open Issues
https://pubs.aip.org/ajp
en-usWed, 01 Nov 2023 00:00:00 GMTMon, 23 Oct 2023 22:46:44 GMTSilverchaireditor@pubs.aip.org/ajpwebmaster@pubs.aip.org/ajpComputational projects with the Landau–Zener problem in the quantum mechanics classroom
https://pubs.aip.org/aapt/ajp/article/91/11/885/2917294/Computational-projects-with-the-Landau-Zener
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">The Landau–Zener problem, where a minimum energy separation is passed with constant rate in a two-state quantum-mechanical system, is an excellent model quantum system for a computational project. It requires a low-level computational effort, but has a number of complex numerical and algorithmic issues that can be resolved through dedicated work. It can be used to teach computational concepts, such as accuracy, discretization, and extrapolation, and it reinforces quantum concepts of time-evolution via a time-ordered product and of extrapolation to infinite time via time-dependent perturbation theory. In addition, we discuss the concept of compression algorithms, which are employed in many advanced quantum computing strategies, and easy to illustrate with the Landau–Zener problem.</span>911188589210.1119/5.0139717https://pubs.aip.org/aapt/ajp/article/91/11/885/2917294/Computational-projects-with-the-Landau-ZenerMultiplicity counting using organic scintillators to distinguish neutron sources: An advanced teaching laboratory
https://pubs.aip.org/aapt/ajp/article/91/11/936/2917291/Multiplicity-counting-using-organic-scintillators
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">In this advanced instructional laboratory, students explore complex detection systems and nondestructive assay techniques used in the field of nuclear physics. After setting up and calibrating a neutron detection system, students carry out timing and energy deposition analyses of radiation signals. Through the timing of prompt fission neutron signals, multiplicity counting is used to carry out a special nuclear material (SNM) nondestructive assay. Our experimental setup is comprised of eight trans-stilbene organic scintillation detectors in a well-counter configuration, and measurements are taken on a spontaneous fission source as well as two (<span style="font-style:italic;">α</span>,n) sources. By comparing each source's measured multiplicity distribution, the resulting measurements of the (<span style="font-style:italic;">α</span>,n) sources can be distinguished from that of the spontaneous fission source. Such comparisons prevent the spoofing, i.e., intentional imitation, of a fission source by an (<span style="font-style:italic;">α</span>,n) neutron source. This instructional laboratory is designed for nuclear engineering and physics students interested in organic scintillators, neutron sources, and nonproliferation radiation measurement techniques.</span>911193694510.1119/5.0139531https://pubs.aip.org/aapt/ajp/article/91/11/936/2917291/Multiplicity-counting-using-organic-scintillatorsOblique angle collisions between three or more billiard balls
https://pubs.aip.org/aapt/ajp/article/91/11/879/2917285/Oblique-angle-collisions-between-three-or-more
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">Oblique angle collisions between two balls or two disks have been addressed by many authors. This paper describes oblique angle collisions between three or four billiard balls. Measurements and calculations are presented for cases where one ball is incident obliquely on two or three identical balls, which are in contact and initially at rest. Conservation laws alone are not sufficient to predict the outcome. Adding information about the impact forces on each ball (and ignoring friction forces) provides good theoretical agreement with experimental results.</span>911187988410.1119/5.0119656https://pubs.aip.org/aapt/ajp/article/91/11/879/2917285/Oblique-angle-collisions-between-three-or-moreForce on a moving object in an ideal quantum gas
https://pubs.aip.org/aapt/ajp/article/91/11/932/2917284/Force-on-a-moving-object-in-an-ideal-quantum-gas
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">We consider a heavy external object moving in an ideal gas of light particles. Collisions with the gas particles transfer momentum to the object, leading to a force that is proportional to the object's velocity but in the opposite direction. In an ideal classical gas at temperature <span style="font-style:italic;">T</span>, the force acting on the object is proportional to T. Quantum statistics causes a deviation from the T-dependence and shows that the force scales with <span style="font-style:italic;">T</span><sup>2</sup> at low temperatures. At <span style="font-style:italic;">T</span> = 0, the force vanishes in a Bose gas but is finite in a Fermi gas.</span>911193293510.1119/5.0127334https://pubs.aip.org/aapt/ajp/article/91/11/932/2917284/Force-on-a-moving-object-in-an-ideal-quantum-gasIn this issue: November 2023
https://pubs.aip.org/aapt/ajp/article/91/11/865/2917283/In-this-issue-November-2023
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">These brief summaries are designed to help readers easily see which articles will be most valuable to them. The online version contains links to the articles.</span>911186586610.1119/5.0177701https://pubs.aip.org/aapt/ajp/article/91/11/865/2917283/In-this-issue-November-2023Galilean relativity and the path integral formalism in quantum mechanics
https://pubs.aip.org/aapt/ajp/article/91/11/893/2917280/Galilean-relativity-and-the-path-integral
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.</span>911189390210.1119/5.0140018https://pubs.aip.org/aapt/ajp/article/91/11/893/2917280/Galilean-relativity-and-the-path-integralElectric field lines of an arbitrarily moving charged particle
https://pubs.aip.org/aapt/ajp/article/91/11/913/2917279/Electric-field-lines-of-an-arbitrarily-moving
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">Electromagnetic fields of relativistic charged particles have a broad frequency spectrum and a sophisticated spatial structure. Field lines offer a visual representation of this spatial structure. In this article, we derive a general set of equations for the field lines of any moving charged particle. The electric field lines are completely determined by the unit vector from the retarding point to the observation point. After proper transformations, the field line equations describe the rotation of this vector with an angular velocity coinciding with Thomas precession. In some cases, including all planar trajectories, the field line equations reduce to linear differential equations with constant coefficients. We present a detailed derivation of these equations and their general analytical solution. We then illustrate this method by constructing field lines for the “figure eight” motion of an electric charge moving under the influence of a plane wave, including complex field lines in three dimensions.</span>911191392210.1119/5.0124544https://pubs.aip.org/aapt/ajp/article/91/11/913/2917279/Electric-field-lines-of-an-arbitrarily-movingLeading quantum correction to the classical free energy
https://pubs.aip.org/aapt/ajp/article/91/11/923/2917278/Leading-quantum-correction-to-the-classical-free
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ 2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box.</span>911192393110.1119/5.0106687https://pubs.aip.org/aapt/ajp/article/91/11/923/2917278/Leading-quantum-correction-to-the-classical-freeAn alternative derivation of propagator for a linear potential
https://pubs.aip.org/aapt/ajp/article/91/11/946/2917277/An-alternative-derivation-of-propagator-for-a
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">In this note, we provide an alternative route to arrive at the quantum wave function propagator for a particle subject to a linear potential. The derivation of this propagator is given as a problem in the monograph of Feynman and Hibbs.<a href="#c1" class="reflinks"><sup>1</sup></a> The construction of the propagator in terms of path integrals is outlined in Ref. <a href="#c2" class="reflinks">2</a> and explicitly given by Holstein<a href="#c3" class="reflinks"><sup>3</sup></a> and by Brown and Zhang.<a href="#c4" class="reflinks"><sup>4</sup></a> In the standard quantum operator formulation, the propagator of a system with Hamiltonian H ̂ is defined as G ( x , t ; x ′ , 0 ) ≡ ⟨ x | e − i H ̂ t / ℏ | x ′ ⟩ (1)in the position representation and K ( p , t ; p ′ , 0 ) ≡ ⟨ p | e − i H ̂ t / ℏ | p ′ ⟩ (2)in the momentum representation. We are interested in the case where, H ̂ = ( p ̂ 2 / 2 m ) − F x ̂ is the Hamiltonian operator for a particle subject to a constant force <span style="font-style:italic;">F</span> in the positive <span style="font-style:italic;">x</span> direction. Robinett<a href="#c5" class="reflinks"><sup>5</sup></a> derived this propagator in both position and momentum representations by explicitly factoring the operator exp ( − i H ̂ t / ℏ ) with the Campbell–Baker–Hausdorff formula. In the monograph,<a href="#c6" class="reflinks"><sup>6</sup></a> Schwinger derived the propagator using variational methods based on the quantum action principle. Another instructive way to solve the linear potential problem is based on the transformation of reference frame, see Refs. <a href="#c7" class="reflinks">7–9</a> and references therein. Vandegrift<a href="#c10" class="reflinks"><sup>10</sup></a> provided a heuristic derivation of the propagator starting from the classical motion of a uniformly accelerating particle in a linear potential where the particle was described by a Gaussian wave packet. The treatment in Ref. <a href="#c10" class="reflinks">10</a> used position representation; here, we will give a complementary derivation in the momentum representation.</span>911194694810.1119/5.0103857https://pubs.aip.org/aapt/ajp/article/91/11/946/2917277/An-alternative-derivation-of-propagator-for-aThe first Global e-Competition on Astronomy and Astrophysics
https://pubs.aip.org/aapt/ajp/article/91/11/867/2917276/The-first-Global-e-Competition-on-Astronomy-and
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">The first global e-competition on astronomy and astrophysics was held online in September–October 2020 as a replacement for the International Olympiad on Astronomy and Astrophysics, which was postponed due to the COVID-19 pandemic. Despite the short time available for organization, 8 weeks, the competition was run successfully, with 325 students from over 42 countries participating with no major issues. The feedback from the participants was positive and reflects the ways in which such events can boost interest in astronomy and astronomy education. With online activities set to become more prevalent in the future, we present an overview of the competition process, the challenges faced, and some of the lessons learned, aiming to contribute to the development of best practices for organizing online competitions.</span>911186787210.1119/5.0121242https://pubs.aip.org/aapt/ajp/article/91/11/867/2917276/The-first-Global-e-Competition-on-Astronomy-andCoupled oscillations of the Wilberforce pendulum unveiled by smartphones
https://pubs.aip.org/aapt/ajp/article/91/11/873/2917275/Coupled-oscillations-of-the-Wilberforce-pendulum
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">The Wilberforce pendulum illustrates important properties of coupled oscillators including normal modes and beat phenomena. When helical spring is attached to a mass to create the Wilberforce pendulum, the longitudinal and torsional oscillations are coupled. A Wilberforce can be constructed simply from a standard laboratory spring, and a smartphone's accelerometer and gyroscope can be used to monitor the oscillations. We show that the resulting time-series data match theoretical predictions, and we share the procedures for observing both normal modes and beats.</span>911187387810.1119/5.0138680https://pubs.aip.org/aapt/ajp/article/91/11/873/2917275/Coupled-oscillations-of-the-Wilberforce-pendulum S -matrices for simple quantum systems
https://pubs.aip.org/aapt/ajp/article/91/11/903/2917274/S-matrices-for-simple-quantum-systems
Wed, 01 Nov 2023 00:00:00 GMT<span class="paragraphSection">Scattering processes are a standard topic covered in introductory courses on quantum mechanics and particle physics. Unfortunately, a full mathematical treatment tends to be overwhelming for undergraduate students. This article introduces some toy models that are easy to comprehend but still contain the essential features of quantum theory. We define a Hilbert space with state vectors and use creation/annihilation operators to construct transition matrices and <span style="font-style:italic;">S</span>-matrices. We show how perturbation theory gives rise to Feynman diagrams and Feynman rules. We also discuss how we can use symmetry and group theory to restrict what interactions are possible.</span>911190391210.1119/5.0078607https://pubs.aip.org/aapt/ajp/article/91/11/903/2917274/S-matrices-for-simple-quantum-systems