Related papers: Upper-division student difficulties with the Dirac…
Students in physics laboratory courses, particularly at the upper division, are often expected to engage in troubleshooting. Although there are numerous ways in which students may proceed when diagnosing a problem, not all approaches are…
We explore the difficulties that advanced undergraduate and graduate students have with non-relativistic quantum mechanics of a single particle in one spatial dimension. To investigate these difficulties we developed a conceptual survey and…
Understanding electron correlation requires solving inseparable Schrodinger equation. In general, inseparable Schr\"odinger equations cannot be solved analytically. So their solutions are obtained numerically. In this paper we investigate…
The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several…
Many undergraduate students of engineering and the exact sciences have difficulty with their mathematics courses due to insufficient proficiency in what we in this paper have termed clear thinking. We believe that this lack of proficiency…
Almost all theories of physics have expressed physical laws by means of differential equations. One can ask: why differential equations? What is special about them? This article addresses these questions and is presented as an inquiry-based…
After reviewing the algebraic derivation of the Doppler factor in the Lienard-Wiechert potentials of an electrically charged point particle, we conclude that the Dirac delta function used in electrodynamics must be the one obeying the weak…
Favorable outcomes from ongoing research at the University of Colorado Boulder on student learning in junior-level electrostatics (E&M I) have led us to extend this work to upper-division electrodynamics (E&M II). We describe here our…
Physics graduate teaching assistants (TAs) are often responsible for grading. Physics education research suggests that grading practices that place the burden of proof for explicating the problem solving process on students can help them…
In this paper, we study a Dirac boundary value problem where the operator is considered with a derivative of order $\alpha \in (0, 1]$, known as the $F^{\alpha}$-derivative. We prove some spectral properties of eigenvalues and…
The Euler-Lagrange equations (EL) are very important in the theoretical description of several physical systems. In this work we have used a simplified form of EL to study one-dimensional motions under the action of a constant force. From…
We present two add-ons for Mathematica for teaching mathematics to undergraduate and high school students. These two applications, M@th Desktop (MD) and M@th Desktop Tools (MDTools), include several palettes and notebooks covering almost…
Learning quantum mechanics is challenging, even for upper-level undergraduate and graduate students. Research-validated interactive tutorials that build on students' prior knowledge can be useful tools to enhance student learning. We have…
A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…
Drawing appropriate diagrams is a useful problem solving heuristic that can transform a problem into a representation that is easier to exploit for solving the problem. A major focus while helping introductory physics students learn problem…
Spectral functions, such as the zeta functions, are widely used in Quantum Field Theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an infinite discrete distribution of point…
We review and extend existing frameworks on modeling to develop a new framework that describes model-based reasoning in upper-division physics labs. Constructing and using models are core scientific practices that have gained significant…
We discuss an investigation exploring students' difficulties with equations involving resistance, capacitance and inductance. We find that introductory physics students have great difficulty understanding, e.g., how the resistance of an…
We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small…
Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a…