Abstract This paper provides a pedagogical overview of the anomalous quantum Hall effect in monolayer graphene subjected to a magnetic field perpendicular to the graphene plane. The energy states of the fundamental charge carriers – electrons and holes – are examined using the Dirac–Weyl framework, showing that their Landau level spectrum depends on the square root of the quantum number n, unlike the uniform spacing found in two-dimensional nonrelativistic charge carriers. A distinctive feature of graphene is the zero-energy Landau level, which lies exactly at the Dirac point and is therefore shared equally between the conduction and valence bands. Its spin–valley degeneracy is the same as that of every other level, but only half of its states are counted on the electron side of charge neutrality. This is what produces the characteristic half-integer quantization of the Hall conductivity, the hallmark of the anomalous quantum Hall effect in graphene. The topological origin of this phenomenon is discussed: the pseudospin structure of massless Dirac fermions imparts a Berry phase of π to the Bloch states when encircling a Dirac point, shifting the Landau level index by ½ compared to the conventional Schrödinger case. The connection between the degeneracy counting argument and the topological description via the Thouless–Kohmoto–Nightingale–den Nijs (TKNN) formula and the Chern number is made explicit, demonstrating that these two approaches are complementary aspects of the same underlying physics.
Abstract Using both WKB approximation methods and complete eigenstate solutions, we examine four standard, textbook level, bound state problems in quantum mechanics. We show how the action integral from the Bohr-Sommerfeld quantization condition, extended to the WKB energy quantization condition, can be expressed in the language of semi-classical and exact quantum results. Relating classical time averaging and quantum expectation value methods, the action integral can be used to connect the average value of speed squared times period across such systems giving 〈v〉 2 n τ n = ˜n(h/m), where ˜n is the WKB quantum number supplemented by the Maslov correction. We then extend this analysis using exact results for an infinite hierarchy of potentials, ones derived from the repeated supersymmetrization of the familiar infinite well, as a more mathematically sophisticated example. These discussions provide a novel pedagogical twist on h as the ‘quantum of action’, useful at several levels of the undergraduate quantum mechanics curriculum
Abstract Entropy is one of the most fundamental concepts in physics and information theory (Claude E. Shannon (1948), \cite{Shannon}), its correct understanding is essential for the study of various physical systems, especially in thermodynamics, statistical mechanics, and in thermal process engineering. This blending of concepts from physics and information theory has made it difficult for many students, even specialists to clearly grasp the power, scope, and proper domain of applicability of the theory, particularly at the undergraduate and graduate levels, mainly due to the abundance of highly artificial theoretical models. In this work, we present a systematic and comprehensive study of the entropy applicability, starting from elementary random systems such as fair (unbiased) and biased dice throws. Specifically, we examine single dice with an arbitrary number of faces, i.e., N=s. We build upon previous studies (\cite{SanchezQ1,SanchezQ2}) that serve as a basis to develop a mathematically consistent analysis, while avoiding ambiguities and unnecessary formal excesses. We compute the corresponding entropy measures for generalized cases involving biased distributions using the probability functions derived in these previous works —additionally the Escher-type distribution for demonstrating its equivalence with the Gibbs–Boltzmann distribution— from which the unbiased cases naturally arise as particular limits. Thus, we provide more significant insight into both the analytical development and the visual interpretation of the resulting entropy metrics.\\ This theoretical framework can be used to introduce advanced undergraduate or postgraduate students to the fundamental characteristics of biased random systems (distributions) and their intrinsic entropy, which are more general than standard cases.
Abstract The electric field of a charged particle in uniform motion is not the same as the Coulomb field of a charged particle at rest. In particular, the field strength is reduced parallel to the line of motion and enhanced perpendicular to the line of motion. While this effect can be understood as a consequence of special relativity, the purpose of this paper is to demonstrate this directly from Faraday’s law. The deviation from the Coulomb field is the result of a Faraday electric field induced by the time-changing magnetic field that accompanies the moving charge. Understanding the effect in this way leads to an infinite series of corrections converging on the exact relativistic field.
Abstract In this article, we work with an optical-mechanical analogy model that exploits the analogy between wave mechanics in quantum theory and the scalar wave treatment of optics. We discuss the application of this model to the evanescent waves and the frustrated total internal reflection (FTIR) of light. Further, we demonstrate that this model can also be used to derive the Goos-Hänchen shift for s-polarised light. This work provides the advanced undergraduate students and instructors of optics and quantum mechanics courses with a platform, where they can apply the methods of the quantum theory to optics, which is not discussed in depth in the standard texts.
Abstract Teaching the equilibrium p – n junction is difficult because several of its key physical quantities—space charge, depletion width, electric field, and electrostatic potential—are not directly visible to students. This paper presents an openly available Python-based teaching approach that combines an electrostatic analogy, simplified analytical relations, and numerical solution of the Poisson equation for introductory undergraduate instruction. The aim is not to replace standard semiconductor theory or laboratory work, but to provide a transparent computational layer that helps students connect equations with physical interpretation. The teaching sequence moves from qualitative reasoning to analytical estimation and then to guided Jupyter-based exploration of field and potential profiles. The paper includes representative lesson formats, classroom tasks, analytical–numerical comparisons, and practical guidance for implementation. All materials are available in a GitHub repository, including ready-to-use notebooks, instructor notes, and student worksheets. The main contribution is pedagogical: the article offers a classroom-ready modelling sequence in which otherwise invisible electrostatic quantities can be visualised, interpreted, and discussed without requiring advanced programming experience. The intended readership is undergraduate physics and electronics teachers seeking a usable teaching resource rather than a research-grade semiconductor simulation study.
Abstract Standard treatments of curvilinear motion decompose acceleration into tangential and normal components, but the resulting components are seldom assembled into an explicit expression for the global direction of the acceleration vector in coordinate-free terms. This tutorial collects, organizes, and works through such an expression for both planar curves and space curves in R³, with worked exercises intended for classroom use. Using the Frenet–Serret frame, curvature κ, and torsion τ, we present four results. First, for a smooth planar curve, the absolute direction of acceleration is assembled into the exact, coordinate-free formula θa(s) = Θ₀ + ∫κ dσ + atan2(κṡ², s̈), whose two terms separate cleanly by what they require of the observer: the tilt φ = atan2(a_N, a_T) is read directly from an onboard accelerometer triad and needs no external input at all, whereas the path orientation Θ(s) requires both an external axis Θ₀ and an external initial speed ṡ₀. Following Galileo and Newton, speed itself is not an intrinsic quantity even in Newtonian mechanics, and we make the resulting three-level hierarchy explicit throughout. Second, for any C² space curve, the acceleration vector is confined to the osculating plane: the binormal component a_B = a·B vanishes identically. Third, the acceleration tilt φ is τ-independent and remains onboard-measurable in three dimensions; the absolute spatial direction of acceleration is â = cos φ T(s) + sin φ N(s), where T, N are determined by integrating both κ and τ. Fourth, setting τ ≡ 0 reduces the three-dimensional formula to the planar formula exactly — the planar result is a strict special case, not an independent result. Worked examples on a circular path and a circular helix confirm consistency with direct Cartesian computation.
Abstract Systems that start farther from their target can, counterintuitively, arrive sooner. Such nonmonotonic relaxation phenomena appear across physics, chemistry, and biology. This paper takes the brachistochrone problem as a mechanical entry point and constructs a teaching analogy model in which the path of descent is freely configurable: two point masses start from different heights and slide without friction to a common endpoint along independently chosen paths. Taking the red ball on a cycloid (the brachistochrone) and the blue ball on an inclined straight line as the canonical case, a rigorous mathematical analysis proves the existence of a parameter interval—which we call the overtaking zone—in which the higher-starting red ball reaches the endpoint before the blue ball, demonstrating the logical structure ‘path optimization leads to temporal reversal’ within classical mechanics. The model does not construct a microscopic mechanism for real thermal relaxation; its pedagogical contribution is to provide a precise, transparent, and interactive mechanical demonstration platform for the concept of path dependence, a fundamental idea in nonequilibrium statistical physics. An interactive simulation tool accompanies the paper, along with a teaching discussion framework.
For two hundred years - ever since Faraday's first conception of the field lines - hearts and minds of students have been pervaded by erroneous and unsubstantiated claims about these well defined mathematical objects. The most prominent misconceptions include: (1) a notion of the field lines as of 'lines of force'; (2) a notion that the field lines coincide with particle trajectories; (3) a notion that a field line density measures a magnitude of a vector field; (4) a notion that the field lines of a divergence-free field always form closed loops. Even the modern day literature systematically perpetuates some of these claims. We compile here the trivial counterexamples to these claims, providing physics instructors with an efficient and effective way of dispelling these misconceptions.
Abstract We present a compact cubical avatar of the su ( 3 ) adjoint (octet) structure. An orthogonal projection of the 3-cube along a body diagonal produces the A 2 root hexagon, while the two axial vertices realize the multiplicity-two zero weight. Motivated by the standard plaquette-holonomy identity from lattice gauge theory, we define an oriented face (plaquette) local rule that computes the nonzero commutators in a Chevalley-like step basis by a square-completion operation. We also record the corresponding sparse 8 × 8 adjoint matrices and show how their sparsity reflects the face-incidence structure of the cube. This note is intended as a mnemonic/pedagogical device for the octet algebra and the baryon-octet weight diagram, not as a new physical model nor as a lattice-QCD algorithm.
Abstract This work combined experiment with a Monte Carlo model to study electron transport in a mercury Franck–Hertz tube under a high-excitation-state measurement, where the accelerating voltage between the cathode and the first grid is swept. The simulated current–voltage curves are in good agreement with the experimental results. The Monte Carlo model resolves the spatial evolution of electron energy distribution, electron density, and electron velocity distribution inside the tube. The results provide a kinetic interpretation of the higher-excitation features in the current–voltage curves and clarify how the underlying transport processes differ from those in the first-excitation measurement. These findings provide a clearer microscopic picture of electron kinetics in the Franck–Hertz experiment.
Abstract We describe the design, construction, and statistical characterization of a hands-on experimental quantum random number generator, using low-cost, accessible materials, that can be used to teach undergraduate students practical, foundational concepts in Quantum Information Science. The base-emitter junction of a PN2222 bipolar junction transistor was reverse-biased at 9 V, and used under avalanche breakdown to generate stochastic, white noise. Components are chosen specifically to ensure the noise from the quantum-mechanical process of avalanche breakdown dominates the signal. This noise is then amplified with an MCP-6022 operational amplifier and digitized through an ADS1115 ADC operating in differential mode. The circuit is interfaced with an Arduino Mega microcontroller and then processed with a Von Neumann whitening algorithm to produce a bitstream that is statistically indistinguishable from an MT19937 pseudo-random baseline. The NIST SP 800–22 tests of Monobit, Serial, Runs, and Longest Run were chosen for their suitability for an undergraduate lab session and were applied to three independent runs; 14 out of 15 test instances yielded a passing p -value. The final extracted bitstream achieved a mean min-entropy of 0.9885 ± 0.0106 bits/bit (mean ± SD, N = 3 ). The limitations of the raw entropy from the physical source were mitigated by a robust post-processing framework. The aim of this paper is to provide an educational experiment for institutions, including those with a lack of resources, to teach Quantum Information Science and Technology to undergraduates and to provide an introduction to the field, along with its challenges.
Abstract In the last decades, empirical investigations have shown the existence of deep and persistent difficulties with teaching/learning the time independent Schrödinger equation and time evolution. At the same time, research has provided evidence in favor of engaging students in modeling activities to help them improve their understanding. However, there is a lack of model-based curriculum materials for teaching the concepts of energy and dynamics in quantum mechanics. To illustrate how modeling could be included in the learning of these topics, we outline an educational proposal in the context of a spins-first approach. As a starting point to introduce the energy concept for atoms and particles, the learning path leverages student knowledge of the Bohr model. The energy level structure of a hydrogen atom with transitions accompanied by the absorption or emission of a photon provides an intuitive basis for activating the modeling of energy, that is organized around the analysis of contrasting cases on the possible results of spin flip transitions in a magnetic field. In this process, students are guided to build a mathematical representation of energy and to investigate its nature up to a derivation of the energy eigenvalue equation. The learning path allows students to compare and contrast the semi-classical Bohr model with the quantum model, as a support to help them build a consistent understanding of the latter. A different set of contrasting cases on the possible results of spin measurements after different amounts of time in the field is used to activate a qualitative modeling process on time evolution, which is then quantitatively examined by analyzing graphs of the state tomography, guiding students to explore the basic features of quantum dynamics up to a derivation of the Schrödinger equation.
Abstract We present a compact experimental setup designed to investigate selected optical properties of rare-earth-doped glasses. Such materials, like doped crystals, are widely used in modern technologies, particularly in lasers and optoelectronic systems. The setup is intended for teaching optical measurements within a general engineering curriculum with an emphasis on materials science and glassy materials. The experiment focuses on a neodymium-doped phosphate glass and is based on optical absorption and optical pumping using a laser diode emitting at the main absorption peak of Nd³⁺ ions in an amorphous matrix. Under continuous pumping conditions, students determine the refractive index and the absorption coefficient as functions of the optical path length and the neodymium concentration and use these measurements to estimate the absorption cross-section with reasonable uncertainty. Using chopped optical pumping, they also measure the excited-state lifetime as a function of the absorbing-ion concentration. Additional investigations include the geometrical characteristics of spontaneous emission and the influence of experimental parameters such as wavelength filtering before the photodiode. The setup integrates PETG 3D-printed components, an Arduinobased control module, and in-house fabricated samples on a compact platform requiring only standard laboratory instruments (laser power meter, photodiode, and oscilloscope). The experiment also introduces students to laser safety considerations associated with a 100 mW continuous-wave near-infrared diode laser. All 3D design files and the data required to reproduce or adapt the setup are freely available for download.
Lattice dynamics provides a microscopic understanding of the thermal, mechanical, and transport properties of crystalline solids; however, undergraduate instruction often remains limited to simplified one-dimensional models. In this work, we present a unified and pedagogically accessible lattice-dynamical framework for real three-dimensional crystals that combines phenomenological force-constant models with first-principles density functional theory (DFT). The de Launay central-angular-force model is employed to compute the phonon dispersions of body-centered-cubic sodium, illustrating the role of lattice symmetry and the necessity of longer-range interactions, while the Clark-Gazis-Wallis three-body model is applied to diamond-structure silicon to elucidate the dominant influence of angular force constants in covalent solids. These classical treatments are benchmarked against DFT phonon calculations obtained through an open-source Python workflow based on ASE, GPAW, and Phonopy and are further validated by comparison with available neutron-scattering data. By presenting classical and ab initio phonon calculations side by side for the same materials, this study clarifies the physical meaning, applicability, and limitations of force-constant models and demonstrates how research-grade lattice-dynamical simulations can be meaningfully integrated into undergraduate instruction using accessible computational tools, enabling students to move beyond idealized textbook models toward realistic materials modeling in contemporary condensed-matter physics. This work is primarily intended for postgraduate (MSc) students, doctoral researchers, and early-career scientists in solid-state physics and materials science, aiming to bridge the gap between simplified textbook treatments of lattice dynamics and modern first-principles computational approaches.
Abstract Immersive virtual reality (IVR) can heighten presence and enable active, embodied interaction in realistic 3D environments, which has been associated with potential benefits in physics education. This article presents a comprehensive review of IVR implementation in physics education at both school and university levels. The analysis included 34 studies indexed in Scopus and ERIC, published between 1 January 2018 and 1 June 2025. Methodologically, this review followed a preferred reporting items for systematic reviews and meta-analyses approach. IVR implementations for physics learning were assessed, topic-aligned activity types were summarized, and associated opportunities and challenges were documented. Findings indicate that IVR can support physics learning by providing complementary visualizations, optimizing cognitive load, enabling haptic learning, saving time, and fostering collaborative inquiry. Conversely, the most frequent potential IVR challenges include discomfort, unreliable interaction, orchestration issues, relatively high costs, and significant extraneous cognitive load.
We study the Fraunhofer diffraction produced by one-dimensional binary gratings based on the Silver Mean sequence, a deterministic aperiodic ordering closely related to the Pell numbers and the silver ratio. Analytical expressions for the diffraction patterns are derived within the scalar approximation, revealing a characteristic redistribution of diffracted energy and the splitting of the first diffraction orders into multiple well-defined maxima. We show that the transverse positions of these maxima are governed by simple combinations of Pell numbers, reflecting the scaling properties of the underlying aperiodic structure. The theoretical predictions are experimentally verified using a simple optical setup based on a spatial light modulator, which can be readily implemented in undergraduate teaching laboratories. These gratings provide an accessible and pedagogically valuable extension of diffraction experiments based on periodic, fractal and Fibonacci-type structures, allowing students to explore the optical consequences of deterministic aperiodic order through direct observation of their diffraction patterns.
Abstract We propose a pedagogical, rationalized MKS-based convention for electromagnetic quantities designed to reduce cognitive load in undergraduate electromagnetism. By setting vacuum constants to ε 0 = μ 0 = 1 / c , we preserve the familiar structure of Maxwell’s equations while making the role of the speed of light explicit. In this convention, electrical units are expressed directly in terms of mechanical units (e.g. [ nuA ] = J / s ), effectively reducing the number of independent base units. A striking pedagogical consequence is that electrical resistance becomes dimensionless, capacitance and inductance acquire units of time, and radiation pressure reduces to | E × B | , greatly simplifying dimensional analysis for circuits and fields. We introduce corresponding non-SI units ( nu -units), provide conversion relations to SI, and demonstrate the potential utility of this system through comparative ‘before/after’ derivations of the wave equation, electromagnetic energy density, radiation pressure, and the Bohr atom. Preliminary empirical support is provided by student attitude surveys administered to N 1 = 46 and N 2 = 39 students in an undergraduate physics course, which showed a statistically significant improvement in the perceived clarity of the wave equation derivation after exposure to the nu-system ( p = 0.005 , Mann–Whitney U test), and a majority preference for the dimensionless-resistance feature.
Cosmic rays provide an important bridge between microscopic and macroscopic physics in undergraduate modern-physics laboratory courses, yet traditional instructional cosmic ray experiments are often limited by considerable cost, low counting rates, and poor reproducibility, making it difficult to obtain stable and reliable results within class time. Based on the FLUKA Monte Carlo simulation framework, we simulated three classic cosmic ray teaching and outreach experiments-ground level secondary particle composition, muon lifetime measurement, and the zenith angle dependence of the muon flux-and found that the results agreed with theoretical expectations while reproducing representative experimental trends. The framework was further applied to the differential energy spectrum of neutrons and the near-vertical differential momentum spectrum of muons, and the resulting spectral features were consistent with representative measurements, demonstrating its extensibility to a broader range of observables. This work indicates that the FLUKA simulation framework enables students to investigate cosmic rays through low cost computational simulations and provides practical guidance for the design of cosmic ray experiments.
Crystallography is a core topic in undergraduate solid-state and materials physics, yet students often find it conceptually difficult due to its abstract, mathematical, and geometric nature. This paper presents a teaching-oriented approach that integrates Rietica software as a learning tool to support students' engagement with fundamental crystallography concepts, including Bragg's law, Miller indices, and basic powder diffraction analysis. The instructional framework was implemented in a fifth-semester undergraduate physics course through a four-session project-based module involving 20 students. Students analyzed real x-ray diffraction data for NdFeO3, starting with phase identification using Match! and then performing Rietveld refinement in Rietica. The student-generated outputs, including lattice parameters, multi-phase identification, and refinement quality indicators such as goodness of fit, were used as the basis for classroom discussion and reflection. This work does not constitute a formal educational experiment; rather, it focuses on the design of learning activities and qualitative pedagogical insights derived from authentic student outputs. The proposed approach is intended to bridge theoretical concepts and practical data interpretation, and suggests a potentially transferable instructional model for introducing crystallography at the undergraduate level.
