Abstract Many textbooks discuss the diamagnetic response of a ``classical atom'' to a small and slowly (adiabatically) applied magnetic field. Here we show that the small-field assumption is not essential, whereas the slow rate of field change is crucial: it preserves the circular nature of the orbit regardless of the applied field magnitude. The latter property follows from the existence of an adiabatic invariant of electron's motion. Orbit circularity allows us to solve the problem for arbitrarily large fields. Resulting diamagnetic response exhibits symmetries that unexpectedly persist beyond the small-field limit, and provides a nice illustration of the range of applicability of Larmor's theorem.
A grounded, zero-thickness hemispherical conductor of radius \(R\) with a point charge \(q\) at the centre of its parent sphere is superficially close to the familiar closed conducting shell, yet the open rim changes both the electrostatics and the numerical analysis: shielding becomes partial, and the induced surface charge is singular at the rim. The problem also admits a classical closed-form solution through Kelvin's treatment of the charged spherical bowl, which we rederive in a form suited to an upper-undergraduate course: \(Q_{\mathrm{ind}}/q=-(\tfrac12+1/\pi)\), the pole density is \(\sigma(0)/\sigma_0=-(\tfrac12+2/\pi)\) with \(\sigma_0=q/(4\pi R^2)\), the rim divergence is \(\sigma/\sigma_0\sim-(2/\pi)/\sqrt{\cos\theta}\), and the residual far-field monopole is exactly \((\tfrac12-1/\pi)q\). We then use the exact solution as the verification target of an axisymmetric boundary-element calculation in which the inverse-square-root edge singularity is extracted analytically by the substitution \(u=\sqrt{\cos\theta}\). The comparison exposes a cautionary result of direct pedagogical value: a mesh sequence whose successive values change at the \(10^{-7}\) level can appear converged while its true error, dominated by the slowly decaying quadrature error of the fixed panel rule, remains several times larger---a discrepancy revealed only by comparison with a refined quadrature rule and the exact solution. We quantify the full error budget---discretisation, panel quadrature, and an independent boundary-potential check---and organise the material as a predict--compute--compare activity in which students must coordinate physical and algorithmic reasoning. Starter and reference codes are provided as supplementary material.
Abstract We present a computational module for upper-level undergraduate quantum mechanics and computational physics courses in which neural networks serve as variational trial wave functions for the Rayleigh–Ritz method. Spatial derivatives are evaluated using automatic differentiation. The infinite square well and harmonic oscillator provide analytically solvable benchmarks, while a quartic anharmonic oscillator illustrates how numerical accuracy can be assessed without a simple analytical solution using network-size convergence, energy variance, and cross-network wave-function overlap. Boundary conditions, decaying envelopes, parity, and orthogonality are incorporated into the trial functions as physical constraints. The examples also demonstrate that excited states require orthogonality to lower states and that a small energy error does not necessarily imply an accurate wave function or node position. The module is intended as a visual and modifiable framework for teaching variational reasoning and numerical error diagnostics.
The order in which the seven International System of Units (SI) base units are listed is often encountered by physics students as a fixed convention, but its historical evolution provides a useful route into deeper questions about dimensions, unit definitions and SI derived unit notation. This article reviews the changing presentation of SI base unit and SI derived unit expressions in the SI Brochure, with the aim of supporting undergraduate teaching of units, dimensions and the revised SI. It traces the presentational sequences adopted from pre-SI practice through successive editions of the SI Brochure, identifying the roles of pedagogy, technical domain-building narratives and, since 2019, definition-led ordering. The article also shows that the ordering of SI base unit symbols in SI derived unit expressions has alternated between following the SI base unit list and reflecting the underlying physics. For reasons of clarity and pedagogical value, the latter approach was reinstated in the 9th edition of the SI Brochure. Finally, the article discusses the tension between conventions that aid human interpretation and forms that support machine readability, using quotient notation and unit names as examples of possible ambiguity.
The technique of coarse-graining is conventionally used in both introductory and more advanced textbooks on statistical mechanics and thermodynamics when discussing a thermodynamic state described by an ensemble approaching equilibrium. In quantum statistical thermodynamics there exists a natural uncertainty-limited phase space volume element, but in the case of classical statistical thermodynamics no such natural phase space volume element exists and coarse-graining by its nature can appear as a rather arbitrary and ad-hoc procedure for students first encountering it. The present work discusses the approach to equilibrium in the ensemble picture, based on observable thermodynamic properties of the system in question. We suggest that a fruitful teaching approach can be based on the perspective that the system has reached thermodynamic equilibrium (distinct from statistical equilibrium) if all its observable macroscopic thermodynamic properties are indistinguishable from those of a canonical ensemble with the same observable macroscopic properties. This differs to an approach based on whether the probability distribution function (specifically the integral of the natural logarithm of the probability distribution) match those of a canonical distribution (which they cannot do in thermally isolated systems, due to the Liouville theorem). These ideas are essentially those of 'weak convergence', normally encountered in more mathematically advanced treatments, but whose basic ideas can be motivated with a simple mathematical outline, and is based on a set of common and reasonable assumptions concerning the evolution of probability distribution functions in a thermally isolated system. The approach may help alleviate student concerns over the use of coarse-graining in classical statistical thermodynamics. A relatively small elevation in the mathematical level of the concepts therefore provides a considerable gain in clarity and coherence of the emergent physical picture, and aligns with previous work by Jaynes, recently highlighted by me, on the definition of entropy in the ensemble picture.
