**October 2023 Issue,**

**Volume 91, No. 10**

**Special issue in celebration of the International Year of Quantum Science and Technology. **

All objects and some questions featured

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.

#### EDITORIAL

In this issue: October 2023 by Joseph C. Amato; Harvey Gould; Jesse Kinder; Raina Olsen; Beth Parks; B. Cameron Reed; Todd Springer; Jan Tobochnik.

#### PAPERS

Why and how to implement worked examples in upper division theoretical physics by Philipp Scheiger; Holger Cartarius; Ronny Nawrodt. DOI: 10.1119/5.0105612

Some of the problem-solving techniques that we teach in upper-level physics require students to follow somewhat formulaic procedures. These techniques include setting up a problem in Lagrangian mechanics, solving boundary-value problems in electrostatics, and finding the time evolution of non-stationary wave functions. For learning to solve these sorts of problems, the authors show that a four-step method of using worked examples can be very beneficial. Supplementary material includes useful background on cognitive load theory and a set of worked examples for Lagranian mechanics.

Damped harmonic oscillator revisited: The fastest route to equilibrium by Karlo Lelas; Nikola Poljak; Dario Jukić. DOI: 10.1119/5.0112573

Just when you thought you fully understood the damped harmonic oscillator, this article may surprise and delight you. Students and instructors will be intimately familiar with the claim that a critically damped harmonic oscillator returns to equilibrium more quickly than all others. While this is true in an idealized and asymptotic sense, the situation is more subtle if experimental resolution is taken into consideration. If we only require the oscillations to drop below a given threshold, a particular underdamped oscillator is more efficient than the critically damped one. This paper discusses this optimal damping coefficient, how to find it, what advantage it offers over the critically damped oscillator, and explores a variety of other fascinating avenues.

An analysis of the large amplitude simple pendulum using Fourier series by Brennen Black; Vetri Vel. DOI: 10.1119/5.0130943

The authors start with an excellent question: since pendulums exhibit periodic motion, why not use the Fourier series to analyze their motion outside the small-amplitude approximation? Undaunted by the realization that normally we need to know the frequency in advance in order to use the Fourier series, they proceed to develop a creative approach that not only leads to accurate approximations but also—in the higher-order corrections—lets instructors introduce students to perturbation analysis.

A magnetic field based on Ampère's force law by Chananya Groner; Timothy M. Minteer; Kirk T. McDonald. DOI: 10.1119/5.0134722

When I teach magnetic fields, currents, and cross products to introductory students, I sometimes find students staring at me, thinking, “Why is she making this so hard? Surely there must be an easier way to formulate these fields and forces!” In this paper, the authors explore a different formulation of magnetic fields that could have been developed when electromagnetism was studied in the 19th century. Students may be disappointed to learn that it's not easier. However, the paper presents an interesting alternate history for the development of electromagnetism, and, in reading that alternate history, readers will gain much information about the actual history. Additionally, instructors who want to create new problems in vector calculus will find many excellent examples in this paper.

Acceptable solutions of the radial Schrödinger equation for a particle in a central potential by J. Etxebarria. DOI: 10.1119/5.0141536

This paper presents a new argument for why the radial part of the wave function cannot diverge at the origin.

Complementarity and entanglement in a simple model of inelastic scattering by David Kordahl. DOI: 10.1119/5.0141389

The interaction of a particle or beam with a harmonic oscillator is treated in three ways: classically, with a quantized harmonic oscillator, and with both the beam and the harmonic oscillator quantized. Since these situations can all be solved without advanced mathematical methods, this interaction provides quantum mechanics students an excellent opportunity to explore both complementarity and entanglement.

Scattering of identical particles by a one-dimensional Dirac delta function barrier potential: The role of statistics by P. R. Berman; Alberto G. Rojo. DOI: 10.1119/5.0089907

The authors present a wave packet analysis of 1-D scattering that illustrates the important role of particle statistics - boson versus fermion - even for noninteracting particles. Two identical particles incident on a delta-function barrier from opposite directions are scattered in the same direction if they are bosons or in opposite directions if they are fermions. The analysis is appropriate for a first course in quantum mechanics. A Mathematica notebook allows readers to explore the model.

Treating disorder in introductory solid state physics by Dunkan Martínez; Yuriko Baba; Francisco Domínguez-Adame. DOI: 10.1119/5.0133701

In 1977, Philip Anderson won the Nobel Prize in physics for his seminal study of electron transport in disordered lattices. Disordered systems exhibit many interesting and important properties, including electron localization, band-splitting, and metal-insulator transitions. However, because of the lack of translational symmetry, the standard theoretical analysis of these materials is quite complex and – up until now – beyond the scope of undergraduate coursework in solid state physics. Starting from the tight-binding approximation, the authors develop the coherent potential approximation and use this tool to find the electron density of states in 1-D binary alloys. This is all accomplished without advanced mathematical techniques, making it highly instructional and suitable for inclusion in introductory solid-state courses.

All objects and some questions by Charles H. Lineweaver; Vihan M. Patel. DOI: 10.1119/5.0150209

This manuscript presents 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. Its “Figures of Everything” are especially detailed, colorful, and well presented. The is valuable not only to instructors but also to professionals because it so strikingly provides an overview of how the Universe has evolved, allowing physicists to see the big picture through its carefully constructed figures in addition to its well written text. A video abstract accompanies the online version of this paper.

#### COMPUTATIONAL PHYSICS

On numerical solutions of the time-dependent Schrödinger equation by Wytse van Dijk. DOI: 10.1119/5.0159866

This paper discusses how an explicit algorithm can be implemented for numerically solving the time-dependent Schrodinger equation to arbitrary levels of precision. Numerical examples of how the algorithm works are given for several systems.

#### ADVANCED TOPICS

Generalized Gaussian integrals with application to the Hubbard–Stratonovich transformation by Krzysztof Byczuk; Paweł Jakubczyk. DOI: 10.1119/5.0141045

Graduate-level courses in statistical and many-body physics make use of Gaussian integrals, especially in the Hubbard-Stratonovich transformation. This paper fills a hole in standard textbooks, providing a careful derivation of the transformation.

#### INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS

An undergraduate physics experiment to measure the frequency-dependent impedance of inductors using an Anderson bridge by Andrew James Murray; Carl Hickman. DOI: 10.1119/5.0148114

lectronics instructors are surely familiar with the Wheatstone bridge, but they may not be familiar with the Anderson bridge that enables measurements of inductance. The authors show how to incorporate this measurement technique in a laboratory class and also show how it can be used to measure and understand nonlinearities in inductors that occur due to skin depth, winding proximity, eddy currents, and core effects.

#### NOTES AND DISCUSSIONS

A note on combined sliding and rolling friction by Rod Cross. DOI: 10.1119/5.0149826

Kinetic friction with sliding objects and the behavior of rolling objects are standard fare for introductory-level dynamics students. It is rare, however, to encounter problems where both phenomena are in play simultaneously. This paper reports video analysis of images of billiard balls projected on a flat felt surface by using a cue to give the balls a small amount of backspin so that they slide for a couple tenths of a second before beginning pure rolling motion. Results are consistent with a model where the coefficient of friction is treated as including two contributions, a rolling component and a part that depends on the acceleration of the contact point of the ball with the surface. Appropriate for introductory and intermediate-level dynamics students.

The emergence of classical mixtures from an entangled quantum state by Mark G. Kuzyk. DOI: 10.1119/5.0063636

Much of the formulation of many-body quantum mechanics was done in the context of closed quantum systems, in which coupling of the system under consideration to its environment is ignored. However, analysis and design of quantum technologies typically require an open quantum systems formalism, which relies on the density matrix. Though this full formalism is clearly a topic for upper-level students, the growing importance of quantum information makes it advisable to introduce key concepts like entanglement in introductory quantum courses. This note suggests a similar introductory pedagogical approach to the density matrix through a simple exercise that shows how apparent classical mixtures arise through entanglement of a system to its environment.

#### Additional Resources