American Journal of Physics Current Issue
https://pubs.aip.org/ajp
en-usSun, 01 Oct 2023 00:00:00 GMTThu, 21 Sep 2023 22:46:50 GMTSilverchaireditor@pubs.aip.org/ajpwebmaster@pubs.aip.org/ajpA note on combined sliding and rolling friction
https://pubs.aip.org/aapt/ajp/article/91/10/855/2912096/A-note-on-combined-sliding-and-rolling-friction
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">The coefficient of sliding friction is usually regarded as a constant for any given surfaces although it is sometimes known to vary with sliding speed. An experiment is described where the coefficient of friction for a billiard ball decreases to a very small value as the sliding speed decreases to zero, after which the ball starts rolling without sliding. A simple theoretical explanation is provided.</span>911085585710.1119/5.0149826https://pubs.aip.org/aapt/ajp/article/91/10/855/2912096/A-note-on-combined-sliding-and-rolling-frictionOn numerical solutions of the time-dependent Schrödinger equation
https://pubs.aip.org/aapt/ajp/article/91/10/826/2911839/On-numerical-solutions-of-the-time-dependent
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.</span>911082683910.1119/5.0159866https://pubs.aip.org/aapt/ajp/article/91/10/826/2911839/On-numerical-solutions-of-the-time-dependentThe emergence of classical mixtures from an entangled quantum state
https://pubs.aip.org/aapt/ajp/article/91/10/858/2911827/The-emergence-of-classical-mixtures-from-an
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">National Science Foundation10.13039/100000001EFRI-ODISSEI:1332271National Sceince FoundationECCS-1128076</span>911085886010.1119/5.0063636https://pubs.aip.org/aapt/ajp/article/91/10/858/2911827/The-emergence-of-classical-mixtures-from-anAcceptable solutions of the radial Schrödinger equation for a particle in a central potential
https://pubs.aip.org/aapt/ajp/article/91/10/792/2911826/Acceptable-solutions-of-the-radial-Schrodinger
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part <span style="font-style:italic;">R</span>(<span style="font-style:italic;">r</span>) of the wave function to be finite at <span style="font-style:italic;">r</span> = 0, showing that if <span style="font-style:italic;">R</span>(0) diverges the complete wave function <span style="font-style:italic;">ψ</span> does not satisfy the full Schrödinger equation. If <span style="font-style:italic;">R</span>(<span style="font-style:italic;">r</span>) is singular, we show that the corresponding <span style="font-style:italic;">ψ</span> follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior <span style="font-style:italic;">R</span> ∝ 1<span style="font-style:italic;">/r</span> near <span style="font-style:italic;">r</span> = 0, which gives rise to a normalizable <span style="font-style:italic;">ψ</span>, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.</span>911079279510.1119/5.0141536https://pubs.aip.org/aapt/ajp/article/91/10/792/2911826/Acceptable-solutions-of-the-radial-SchrodingerAn undergraduate physics experiment to measure the frequency-dependent impedance of inductors using an Anderson bridge
https://pubs.aip.org/aapt/ajp/article/91/10/847/2911825/An-undergraduate-physics-experiment-to-measure-the
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">One of the most accurate ways to measure the impedance of an electrical component is to place it in a bridge that is then balanced. The most familiar bridge in an undergraduate laboratory is the Wheatstone bridge, which can measure resistance to high precision. Other types are, however, required for reactive components. This paper describes the use of Anderson's bridge to measure inductance, allowing both the inductance and resistance of different inductors to be determined. The inductors are analysed with different cores: perspex, copper, and steel. Models for the inductance that include the effect of skin depth, winding proximity, eddy currents, and core effects are introduced and compared to measurements in the frequency range from 100 Hz to 100 kHz.</span>911084785410.1119/5.0148114https://pubs.aip.org/aapt/ajp/article/91/10/847/2911825/An-undergraduate-physics-experiment-to-measure-theA magnetic field based on Ampère's force law
https://pubs.aip.org/aapt/ajp/article/91/10/783/2911824/A-magnetic-field-based-on-Ampere-s-force-law
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">Ampère's force law for steady currents was not historically associated with a magnetic field, but it could have been. A magnetic field, inspired by work of Helmholtz in 1870, can be defined such that the double-differential form of Ampère's force law is a function of a double-differential of this field. We call this field the Ampère–Weber field, B, and show that its divergence is zero everywhere, as is that of the usual, but different, magnetic field B of Maxwellian electrodynamics. The curl of the Ampère–Weber field is nonzero everywhere in static examples, in contrast to that of the usual magnetic field B. We illustrate the field B for three examples, which exhibit patterns of field lines quite different from those of the usual magnetic field. As the Ampère–Weber field is based on Ampère's force law for steady currents, it does not extrapolate well to the Lorentz force on a moving charge in a magnetic field. That is, the Ampère–Weber field B, like Ampère's force law, is more of a curiosity than a viable alternative to the usual magnetic field B. If the Ampère–Weber field had been invented in the mid-1800s, it would have been a distraction more than a step toward a generally valid electromagnetic field theory.</span>911078379110.1119/5.0134722https://pubs.aip.org/aapt/ajp/article/91/10/783/2911824/A-magnetic-field-based-on-Ampere-s-force-lawDamped harmonic oscillator revisited: The fastest route to equilibrium
https://pubs.aip.org/aapt/ajp/article/91/10/767/2911823/Damped-harmonic-oscillator-revisited-The-fastest
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial condition, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We experimentally confirm some of our findings.</span>911076777510.1119/5.0112573https://pubs.aip.org/aapt/ajp/article/91/10/767/2911823/Damped-harmonic-oscillator-revisited-The-fastestAll objects and some questions
https://pubs.aip.org/aapt/ajp/article/91/10/819/2911822/All-objects-and-some-questions
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">We present an overview of the thermal history of the Universe and the sequence of objects (e.g., protons, planets, and galaxies) that condensed out of the background as the Universe expanded and cooled. We plot (i) the density and temperature of the Universe as a function of time and (ii) the masses and sizes of all objects in the Universe. These comprehensive pedagogical plots draw attention to the triangular regions forbidden by general relativity and quantum uncertainty and help navigate the relationship between gravity and quantum mechanics. How can we interpret their intersection at the smallest possible objects: Planck-mass black holes (“instantons”)? Does their Planck density and Planck temperature make them good candidates for the initial conditions of the Universe? Our plot of all objects also seems to suggest that the Universe is a black hole. We explain how this depends on the unlikely assumption that our Universe is surrounded by zero density Minkowski space.</span>911081982510.1119/5.0150209https://pubs.aip.org/aapt/ajp/article/91/10/819/2911822/All-objects-and-some-questionsAn analysis of the large amplitude simple pendulum using Fourier series
https://pubs.aip.org/aapt/ajp/article/91/10/776/2911774/An-analysis-of-the-large-amplitude-simple-pendulum
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">The motion of a pendulum is derived using Fourier series and perturbation analysis at levels appropriate for undergraduate physics students. Instead of using the elliptic integral of the first kind, higher order terms of the Taylor-expanded differential equation are considered, leading to increasingly accurate corrections to the period in terms of a single expansion parameter. The relation between the expansion parameter and the initial conditions is not fixed, allowing many solutions to the motion in terms of the expansion parameter but a unique solution in terms of the initial conditions.</span>911077678210.1119/5.0130943https://pubs.aip.org/aapt/ajp/article/91/10/776/2911774/An-analysis-of-the-large-amplitude-simple-pendulumComplementarity and entanglement in a simple model of inelastic scattering
https://pubs.aip.org/aapt/ajp/article/91/10/796/2911773/Complementarity-and-entanglement-in-a-simple-model
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.</span>911079680410.1119/5.0141389https://pubs.aip.org/aapt/ajp/article/91/10/796/2911773/Complementarity-and-entanglement-in-a-simple-modelTreating disorder in introductory solid state physics
https://pubs.aip.org/aapt/ajp/article/91/10/812/2911772/Treating-disorder-in-introductory-solid-state
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.</span>911081281810.1119/5.0133701https://pubs.aip.org/aapt/ajp/article/91/10/812/2911772/Treating-disorder-in-introductory-solid-stateSpecial issue in celebration of the International Year of Quantum Science and Technology
https://pubs.aip.org/aapt/ajp/article/91/10/763/2911771/Special-issue-in-celebration-of-the-International
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">In 2025, we'll celebrate the one hundredth anniversary of the development of quantum mechanics, and the United Nations is working towards a declaration of 2025 as the International Year of Quantum Science and Technology (IYQST). The <span style="font-style:italic;">American Journal of Physics</span> will join the celebration with a special issue to kick off the year in January 2025.</span>911076376310.1119/5.0173872https://pubs.aip.org/aapt/ajp/article/91/10/763/2911771/Special-issue-in-celebration-of-the-InternationalGeneralized Gaussian integrals with application to the Hubbard–Stratonovich transformation
https://pubs.aip.org/aapt/ajp/article/91/10/840/2911770/Generalized-Gaussian-integrals-with-application-to
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">We analyze a variety of Gaussian integrals with the aim of revisiting the derivation of the Hubbard–Stratonovich transformation as given in standard graduate-level textbooks and provide an overview of its applications. We pinpoint problematic steps in the usual discussions and propose careful derivations of the Hubbard–Stratonovich identity pertinent to a variety of situations relevant to statistical physics and quantum field theory. These derivations are based on direct use of either a resolution identity or a series expansion. A few homework problems for students are suggested.</span>911084084610.1119/5.0141045https://pubs.aip.org/aapt/ajp/article/91/10/840/2911770/Generalized-Gaussian-integrals-with-application-toScattering of identical particles by a one-dimensional Dirac delta function barrier potential: The role of statistics
https://pubs.aip.org/aapt/ajp/article/91/10/805/2911769/Scattering-of-identical-particles-by-a-one
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.</span>911080581110.1119/5.0089907https://pubs.aip.org/aapt/ajp/article/91/10/805/2911769/Scattering-of-identical-particles-by-a-oneWhy and how to implement worked examples in upper division theoretical physics
https://pubs.aip.org/aapt/ajp/article/91/10/764/2911768/Why-and-how-to-implement-worked-examples-in-upper
Sun, 01 Oct 2023 00:00:00 GMT<span class="paragraphSection">Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and documented in many research projects. In this work, we propose a new four-step approach for teaching with worked examples that includes writing explanations and finding and correcting errors. This teaching method can even be implemented in courses in which homework performance constitutes part of the grading system. This four-step approach is illustrated in the context of Lagrangian mechanics, which is ideal for the application of worked examples due to its universal approach to solve problems.</span>911076476610.1119/5.0105612https://pubs.aip.org/aapt/ajp/article/91/10/764/2911768/Why-and-how-to-implement-worked-examples-in-upperIn this issue: October 2023
https://pubs.aip.org/aapt/ajp/article/91/10/761/2911765/In-this-issue-October-2023
Sun, 01 Oct 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>911076176210.1119/5.0173606https://pubs.aip.org/aapt/ajp/article/91/10/761/2911765/In-this-issue-October-2023