Experimentally determining the isotope shift in atomic spectra leads students beyond their naive understanding of atomic structure and can introduce them to the basics of optical metrology. Standard undergraduate laboratory experiments for the isotope shift on the principal D1 and D2 transitions in rubidium typically use saturated absorption spectroscopy. We show that using a different atomic state, namely the 5D52 level, significantly simplifies the data collection, analysis, and interpretation in comparison to the standard experiments. With two low-bandwidth photodiodes and a Fabry-Perot cavity, the isotope shift is measured to be 162.8 +/- 0.9(stat +/- 1syst) MHz. This result guides students to the necessity of including an similar to 30 MHz (many-body electronic) specific mass shift, in addition to the contributions of the nucleus's countermotion and the finite nuclear charge radius.
Heron's fountain looks like a simple trick: water spurts upward as if the device could run forever. In reality, it is a showcase of several physical phenomena at work—gravity, air compression, viscosity, turbulence, and energy dissipation all intertwined. Surprisingly, this classic demonstration has rarely been treated beyond a qualitative level. In this work, we build a consistent theoretical framework, extending Bernoulli's equation to include viscous effects and clarifying the limits of its applicability within the system. We complement the model with experiments that allow a direct comparison between predictions and data. The result is a new perspective on a two-thousand-year-old invention, in which a familiar classroom curiosity becomes a precise tool to explore fundamental fluid mechanics. From an educational standpoint, the system also provides a flexible platform for laboratory or classroom activities, suitable for advanced undergraduate and early graduate students, and well-suited to illustrate hydrostatics, viscous flow, and the interplay between theory and experiment.
Editor's Note: This paper is based on the Richtmyer Lecture Award given by the author at the AAPT Winter 2026 meeting.
Editor's Note: This paper is the text of a plenary talk given by the author at the AAPT Winter 2026 meeting when she accepted the Doc Brown Futures Award.
A firefly flash contains roughly 108- 1011 photons-far fewer than the 1013- 1014 photons implied by Coblentz's 1912 report of 1/50-1/400 candlepower for Photinus pyralis. We trace this discrepancy to selective citation of the upper end of Coblentz's range and to systematic biases in early visual photometry. We derive a theoretical bound from luciferase abundance and quantum yield. We also measure flash brightness directly with a lux meter and reanalyze two historical datasets. These independent lines of evidence all fall well below the historical candlepower values. The error persisted because modern bioluminescence research often reports quantum yields and relative intensities; reconstructing absolute photons per flash also requires in vivo substrate turnover or measurement geometry, so the comparison with early photometry was rarely made directly.
Viscosity describes how easily a liquid can flow under an applied stress. Dimensionally, it is the product of an elastic modulus and a timescale. The significance of these quantities in determining the viscosity of a liquid can be seen after considering a simple model for liquid flow, in which the flow results from the instantaneous storage of elastic energy and its subsequent dissipation. The modulus then corresponds to the short-time elastic modulus and the timescale is that associated with particle rearrangements. Interestingly, applying conservation of energy, as introduced in continuum mechanics, naturally provides the basis for this way to think about liquid flow. Additionally, the result naturally connects to viscoelastic behavior, emphasizing its applicability to liquids in general, and opening up interesting possibilities to expand the contents often taught in continuum mechanics courses.
Thomson's atomic model is commonly presented in textbooks as an oversimplified and largely qualitative “plum pudding” picture, obscuring its original conceptual and mathematical structure. In this paper, we present a historically grounded reconstruction of Thomson's atom based on excerpts from original papers, showing how it was a dynamically constrained model grounded in classical mechanics and electrodynamics, explicitly addressing atomic structure, stability, and chemical properties. Using this historical example, we discuss key features of scientific modeling, including explanatory scope, empirical constraints, the epistemic role of mathematics, and conceptual continuity across subsequent models, showing that Thomson's atom provides an effective case study for promoting critical reflection on modeling and constitutes a step toward a gradual understanding of the nature of science in the teaching of physics.
Benford’s law describes the nonuniform distribution of leading digits in many naturally occurring datasets, where smaller digits appear more frequently than larger ones. Although this empirical law has been widely discussed in areas such as economics, geophysics, and demographic statistics, its relevance to datasets of physical quantities has received comparatively little attention. In this work, I examine the leading-digit statistics of superconducting critical temperatures (Tc) for representative superconducting materials spanning more than two orders of magnitude. The observed digit distribution is broadly consistent with Benford’s law. I further examine the distribution of the logarithmic mantissa of Tc, which provides an alternative diagnostic for Benford behavior. The results illustrate that discovery-driven materials datasets that span wide dynamic ranges naturally exhibit statistical structures similar to those described by Benford’s law.
Textbook solutions for the harmonic oscillator and the hydrogen atom using the Sommerfeld polynomial method often appear tedious and formidable to students. However, the differential operators acting on the reduced wave functions possess a convenient algebraic property: they preserve the subspace of polynomials of degree at most k. This observation makes the derivation of energy eigenvalues transparent and direct, without the need for the full power series recurrence relations.
Editor's Note: The driven disk is a simple and familiar chaotic system. Wedged between two motorized rollers and confined to the vertical plane, the disk is driven upward. The result is a chaotic mix of spinning and pendulum motion that will surprise and delight experimenters. The new model presented here is accessible to undergraduate physics students in courses on classical mechanics and chaotic systems.
Standard introductory courses in thermodynamics typically introduce the ideal gas equation of state as an empirical postulate or a result of kinetic theory. In this work, we show an alternative way of deriving the equation pV∝NT from general postulates. By assuming only the functional independence of internal energy from volume and enthalpy from pressure under isothermal conditions, or, equivalently, the opposite conditions, we show how the ideal gas law emerges as a necessary consequence of these energy-based symmetries. This serves a dual purpose: it acts as a rigorous exercise in applying thermodynamic relations and multivariable calculus, while simultaneously providing an alternative conceptual perspective on the foundational principles of thermodynamics.
Band formation in periodic media is a central topic in solid-state physics and is typically introduced using Bloch's theorem as an eigenvalue problem for infinite periodic systems. Although elegant, this approach assumes an idealized infinite lattice, shifts attention away from real-space wave dynamics, and presents band structures as static outcomes rather than emergent features of wave propagation. As a result, students often do not connect band gaps with familiar phenomena such as reflection, transmission, and interference, creating a disconnect between formal band theory and observable wave behavior. We address this problem by discussing how band gaps occur from wave propagation in finite periodic systems. A broadband wavepacket is launched into a layered medium using a staggered-grid finite-difference space-time algorithm, and the transmission spectra are used to identify band gaps. The Bloch dispersion relation and the exponential decay of the wave field within band gap regions are extracted, thus establishing a connection between real-space wave propagation and the Bloch description of periodic media.
We present an operator-based derivation of Bloch's theorem. In this approach, we first construct an explicit operator form of the Hamiltonian by writing the potential part in terms of an infinite sum of the momentum-translation operators. Next, we prove that it commutes with the position-translation operator corresponding to a lattice translation, using the exponential re-ordering identity. We then build on Merzbacher's treatment of simultaneous diagonalization of commuting operators to reproduce Kittel's central equation and also to derive a new formula for the simultaneous eigenket corresponding to a specific energy eigenvalue. Moreover, through the Born rule, we provide a precise account of the position and momentum probability distributions for Bloch states, and we examine the concept of crystal momentum. Finally, we apply our findings to the Kronig–Penney model, where we numerically depict the band structure and illustrate the delocalized nature of the single electron.
