teaching
A collection of pedagogical articles, essays, notes, and philosophical reflections
Classical Electrodynamics
These papers explore alternative mathematical and pedagogical routes through classical electrodynamics, with particular emphasis on Helmholtz theorems, causality, retarded fields and electromagnetic potentials.
The Helmholtz theorem and retarded fields
An examination of how the Helmholtz theorem can be extended from static vector fields to time-dependent electromagnetic fields. The paper distinguishes instantaneous and causal formulations and shows how suitable generalisations of the theorem lead naturally to the retarded fields of Maxwell's equations.
Ricardo Heras, European Journal of Physics, 37, 065204 (2016)
Alternative routes to the retarded potentials
A comparison of two routes to the retarded electromagnetic potentials: the standard Lorenz-gauge construction and an alternative derivation based on the retarded Helmholtz theorem. The discussion also contrasts retarded and instantaneous potentials and extends the construction to electrodynamics with magnetic monopoles.
Ricardo Heras, European Journal of Physics, 38, 055203 (2017)
Helmholtz's theorem for two retarded fields and its application to Maxwell's equations
A two-field extension of the Helmholtz theorem in which a pair of retarded vector fields is determined by their divergences and coupled curls. Applied to Maxwell's equations, the theorem provides a direct route from the field equations to the retarded electric and magnetic fields.
José A. Heras and Ricardo Heras, European Journal of Physics, 41, 035201 (2020)
Special Relativity
These papers present elementary routes to the Lorentz transformations, emphasising the geometrical structure of spacetime and the special role played by the wave equation and the d'Alembert operator.
Lorentz transformations and the wave equation
A compact derivation of the Lorentz transformations obtained by demanding form invariance of the d'Alembert operator between inertial frames. The construction provides an alternative introduction to special relativity in which spacetime symmetry emerges directly from the structure of the wave equation.
Ricardo Heras, European Journal of Physics, 37, 025603 (2016)
Four easy routes to the Lorentz transformations: addendum to 'Lorentz transformations and the wave equation'
Four accessible derivations of the Lorentz transformations are compared. Two begin with invariance of the Minkowski interval and two with invariance of the d'Alembert operator, illustrating how apparently different starting points encode the same spacetime symmetry.
Ricardo Heras, European Journal of Physics, 38, 019401 (2017)
Quantum Mechanics
This work uses the Dirac quantisation condition as an entry point into some of the deepest connections between quantum mechanics, electromagnetism, gauge invariance and magnetic monopoles.
Dirac quantisation condition: a comprehensive review
A pedagogical review of the Dirac quantisation condition and its underlying physics. Gauge invariance, singular potentials, single-valued wave functions, the Dirac string and electromagnetic angular momentum are developed through several quantum-mechanical and semiclassical derivations of charge quantisation.
Ricardo Heras, Contemporary Physics, 59, 331–355 (2018)
History of Physics
These papers revisit episodes in the conceptual development of electromagnetism and special relativity, particularly Feynman's approach to electromagnetic potentials and the early role of the wave equation in the emergence of spacetime transformations.
On Feynman's handwritten notes on electromagnetism and the idea of introducing potentials before fields
An exploration of Feynman's 1963 handwritten notes on an alternative formulation of electrodynamics. Motivated by his idea of introducing electromagnetic potentials before the fields, the paper develops a route from charge conservation and causality to the retarded potentials and ultimately to Maxwell's equations.
José A. Heras and Ricardo Heras, European Journal of Physics, 41, 035202 (2020)
Reply to comments 'On Feynman's handwritten notes on electromagnetism and the idea of introducing potentials before fields'
A response to comments on the interpretation and implementation of Feynman's handwritten electrodynamics notes. The discussion clarifies the conceptual motivation for introducing potentials before fields and addresses objections to the proposed pedagogical construction.
José A. Heras and Ricardo Heras, European Journal of Physics, 42, 018002 (2021)
A review of Voigt's transformations in the framework of special relativity
A historical and conceptual review of Woldemar Voigt's 1887 transformations, obtained in connection with the Doppler effect and the wave equation. Their relation to the later Lorentz transformations is examined through the conformal covariance of the wave equation.
Ricardo Heras, arXiv:1411.2559 (2014)
The wave equation in the birth of spacetime symmetries
A historical discussion of the role played by the wave equation in the development of spacetime transformations. The Voigt and Lorentz transformations are compared through conformal covariance and invariance of the d'Alembert operator, revealing their close mathematical and historical relationship.
Ricardo Heras, arXiv:1407.3425 (2014)
Essays and Philosophical Reflections
These essays reflect on independent thought, creativity and the teaching of physics, and on the intellectual conditions that allow students to engage with physics as an exploration of nature rather than simply as a collection of techniques to be mastered.
Individualism: The legacy of great physicists
An essay on the role of independent thought in physics and its relationship with curiosity, doubt, imagination and creativity. Drawing on the examples of major physicists, it argues that fruitful collaboration is strengthened rather than weakened by the intellectual independence of the individuals involved.
Ricardo Heras, Physics Today, online (2013)
Commentary: How to teach me physics: Tradition is not always a virtue
A student perspective on physics education that questions an excessive reliance on rote learning, standard exercises and examinations. The essay argues instead for a form of education centred on free inquiry, independent reasoning and what Feynman described as learning through creation.
Ricardo Heras, Physics Today, 70(3), 10 (2017)
Readers weigh in on how to teach physics
A continuation of the discussion on physics education, defending the importance of intuition and creativity alongside technical practice. The response argues for physics departments to make greater room for independent inquiry, original thinking and creative approaches to learning.
Ricardo Heras, Physics Today, 70(11), 11 (2017)