An overview of classical scattering by a Lennard-Jones potentialKroetz, Tiago
2025 European Journal of Physics
doi: 10.1088/1361-6404/adecce
We present a numerical and conceptual overview of classical scattering under a Lennard-Jones potential. All results were obtained by direct integration of particle trajectories in the two-dimensional plane for varying impact parameters and energies. From these trajectories, we compute deflection functions, identify critical values associated with unstable circular orbits, and visualize non-monotonic behavior in the angular deflection. We construct color maps representing the deflection angle and interaction delay time as functions of energy and impact parameter, and determine the differential cross section from the deflection function, accounting for multi-valued contributions. These results are displayed both as individual plots and as a full color map in the energy-angle plane, providing a detailed picture of the scattering behavior in this central potential.
Operator-based derivation of the wavefunctions of the Morse potentialLi, Jianhao; Zgid, Dominika; Freericks, James K
2025 European Journal of Physics
doi: 10.1088/1361-6404/adeee3
The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
Theoretical and experimental analysis of stable equilibria of floating cylindersKrehbiel, Joel D
2025 European Journal of Physics
doi: 10.1088/1361-6404/adf7e7
In a typical physics course, students determine whether an object floats based on its density relative to that of the fluid. By discussing the orientation of the floating object, students can gain insights into the differences between equilibrium and stability. Here we show how a hollow cylindrical shell with radius R, and height H floats. The answer depends on H/R and the relative density of the cylinder to the density of the fluid: ρ. Assuming that ρ = 0.5, we identify stable equilibria and show what values of H/R produce stable equilibrium that are horizontal, vertical, or angled. The theory agrees well with experimental data from 3D printed cylinders. The experimental portion of this project is easily completed by introductory physics students and shows good agreement with the theoretical angle of stability for solid and hollow cylinders, suggesting a simple way to illustrate the stability of various objects.
An experiment to compare the thermal performance of two thermal insulating materials for domestic heating applicationsAbedigamba, OP; Obia, P; Mawire, A; Rukaaya, M; Okullo, M
2025 European Journal of Physics
doi: 10.1088/1361-6404/adf306
Thermal insulating materials are very essential in thermal energy storage. There are both imported and locally/homemade thermal insulation materials. However, the imported thermal insulation materials are expensive compared to the locally made ones due to the high importation cost and taxes imposed on them. This paper aims to compare two thermal insulators for domestic solar heating applications. The two insulators are a wonderbag (imported from South Africa) and a hay basket (locally made in Uganda). Two identical cooking pots with a capacity of 1 L each are separately placed on identical photovoltaic direct current cookers inside a wonderbag and hay basket, respectively. Water heating experiments are performed in the morning and noon. Results show that when a 0.5 kg water load is used, a higher maximum temperature of 96.3 °C is achieved by the water in a wonderbag insulator compared to 94.8 °C achieved for a hay basket, within 24 min of the experiments during solar noon. A slightly higher water heating efficiency of 3.9% is shown when in the wonderbag compared to that of 3.7% attained when the hay basket is used. No significant difference in the maximum water load temperatures and heating efficiencies are observed when using a wonderbag and hay basket as thermal insulating materials for domestic solar cooking applications during solar noon. Longer heating times are taken during morning experiments (32–34 min) than during noon experiment (26–30 min). The work demonstrates that a cheaply available and locally made heat retention vessel (hay basket) can substitute the imported and expensive heat retention vessel (wonderbag). The experimental setup can be used to introduce the ideas of heat retention, solar cooking, and photovoltaics to senior undergraduate students pursuing courses in thermal physics and semiconductor physics, respectively.
Relativistic length evolution and breakdown of rigidity: a Minkowski diagram analysisLee, Jinho
2025 European Journal of Physics
doi: 10.1088/1361-6404/adf8a4
We analyze the length evolution of a uniformly accelerated rod from the perspective of an inertial frame moving at a constant velocity. Using Minkowski diagrams and Lorentz transformations, we show that a rod which appears rigid in one frame does not retain that rigidity in another. In the asymptotic limit, the rod’s length approaches a constant proportional to the relativistic Doppler factor. These results underscore the breakdown of classical rigidity under relativistic conditions and offer a pedagogical framework for exploring relativistic kinematics in non-inertial systems.
Albert Einstein, Heike Kamerlingh Onnes and the discovery of superconductivityMaas, Ad; Kox, A J
2025 European Journal of Physics
doi: 10.1088/1361-6404/adeb11
In this paper we interpret and contextualize three newly discovered letters from 1911 by Albert Einstein to Heike Kamerlingh Onnes. They shed new light on their mutual relationship, and on how their views were underpinned by a shared, atomistic view on the constitution and behaviour of matter. They reveal how the discovery of superconductivity in this period by Kamerlingh Onnes, was ‘missed’ by Einstein. While Kamerlingh Onnes did the first observations of superconductive behaviour, Einstein tried to persuade him to pursue other experiments, which however only would have prevented Kamerlingh Onnes from making his monumental discovery. Interesting about the experiments Einstein proposed is that they were aimed at providing evidence for his still controversial views on quantum physics.
Hidden golden ratio in two coupled harmonic oscillatorsDe Luca, R
2025 European Journal of Physics
doi: 10.1088/1361-6404/adebc2
The equations of motion for a simple mechanical system, consisting of a pendulum of length land mass mcoupled to a harmonic oscillator formed by an ideal spring and a body of mass m, are derived and solved, using Newtonian mechanics. Interestingly, when the natural frequencies of the two harmonic oscillators are set to the same value ωO, the eigenfrequencies of the combined system are found to be ωOscaled by the golden ratio 5+12and by its reciprocal 5−12. This intriguing relationship between the coupled oscillators and the golden ratio was uncovered serendipitously.
The science of Lato–Lato: tips for successYe, Chengzhi; Yan, Qin; Liu, Sumei; Dong, Zhaoyang; Wang, Mengxiang; Zhang, Kaichao
2025 European Journal of Physics
doi: 10.1088/1361-6404/ade37e
Lato–Lato, a simple toy consisting of two small balls, embodies a wealth of physical principles in its motion dynamics. This paper begins with the fundamental principles of pendulum motion and extends to the nonlinear system of the Lato–Lato. Utilizing Lagrangian mechanics, we explore the conservation of momentum, energy conversion, and resonance phenomena during the oscillation of the Lato–Lato. The differential equations governing the motion of the Lato–Lato are solved, and the Mathieu equation is employed to determine the frequency range for resonance occurrence. Furthermore, the trajectory and swing angle of the Lato–Lato are analyzed. Based on theoretical analysis, this paper proposes a series of recommendations for optimizing the manipulation techniques of the Lato–Lato, including the selection of appropriate Lato–Lato models, control of force and rhythm. The research findings indicate that understanding the scientific principles behind the Lato–Lato, combined with effective practice methods, can significantly enhance manipulation skills, providing a theoretical foundation and practical guidance for the popularization and development of the Lato–Lato.
The Dirac equation: historical context, comparisons with the Schrödinger and Klein–Gordon equations, and elementary consequencesTsutsui, Thiago T; Silva, Edilberto O; M de Castro, Antonio S; Andrade, Fabiano M
2025 European Journal of Physics
doi: 10.1088/1361-6404/adf9a5
This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein–Gordon equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free particle dynamics in Dirac’s model, revealing the emergence of solutions with negative energy. In this discussion, we examine the Dirac Sea Hypothesis and explore the inherent helicity of the solutions. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. Through the Foldy–Wouthuysen transformation, we reveal how the equation incorporates spin–orbit interaction and other relativistic effects, ultimately leading to the hydrogen fine structure. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the Klein–Gordon and Dirac equations. Designed for undergraduate students with an interest in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.