We present an extremely simple derivation of the general relativistic periastron advance for Schwarzschild geometry. The method lends itself to iteration, and so in principle may achieve an arbitrary level of accuracy with relative ease. Our technique involves no analysis of differential equations beyond that of a simple harmonic oscillator. The practical utility of the calculation is limited by its neglect of frame-dragging rotational effects.

We provide an overview of Monte Carlo algorithms based on Markovian stochastic dynamics of interacting and reacting many-particle systems not in thermal equilibrium. These agent-based simulations are an effective way of introducing students to current research without requiring much prior knowledge or experience. By starting from the direct visualization of the data, students can gain immediate insight into emerging macroscopic features of a complex system and subsequently apply more sophisticated data analysis to quantitatively characterize its rich dynamical properties, both in the stationary and transient regimes. We utilize simulations of reaction–diffusion systems, stochastic models for population dynamics and epidemic spreading, to exemplify how interdisciplinary computational research can be effectively utilized in bottom-up undergraduate and graduate education through learning by doing. We also give helpful hints for the practical implementation of Monte Carlo algorithms, provide sample codes, explain some typical data analysis tools, and describe various potential error sources, pitfalls, and tips for avoiding them.

The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

We have investigated the probabilities of earning high (top-quartile) and low (bottom-quartile) course grades in introductory university physics courses for students in two different groups: one, those who scored in the top quartile of their class on one of three diagnostic pretests, and the other composed of those who scored in the bottom quartile on the same test. The tests employed were the Force Concept Inventory (a physics concept test), the Lawson Test of Scientific Reasoning, and a newly developed mathematics test that includes only pre-college mathematics questions; all pretests were administered before or near the beginning of the course. Our investigation includes over 2000 students enrolled in 31 introductory physics classes taught by eight instructors at four universities. We found with 97% consistency that top-quartile scorers on any of the pretests were more likely to get high (top-quartile) grades and less likely to get low (bottom-quartile) grades than were bottom-quartile scorers on the same pretest. Top-quartile scorers on the pretests were, on average, four to six times as likely to receive high grades, and one-third to one-fifth as likely to receive low grades, compared to bottom-quartile scorers on the same pretests. These results are consistent with mathematical models of empirical data published by Salehi et al. [Phys. Rev. Phys. Educ. Res. 15(2), 020114 (2019)] and with their cautions regarding the potentially serious implications of these findings for the careers of poorly prepared college physics students.

Infinite sheets/slabs are one of three geometries, for which the electric field can be calculated using Gauss's law. Frequently, only symmetrical charge distributions are considered, preventing practical applications including p–n junctions and multiple conducting sheets from being introduced in an introductory electromagnetic course. Students also struggle in understanding the requirements for applying Gauss law for calculating electric fields. A common question that confuses students is whether the calculated electric field in a problem corresponds to the enclosed charge within the Gaussian surface or all charge in the distribution. We show that the electric field calculations for infinite sheet/slab charge distributions using the integral form of Gauss's law can be extended in a classroom discussion to asymmetric charges without using superposition, provided the charge is uniform in the two infinite dimensions. These discussions enrich the application of Gauss's law in practical devices and help students develop improved understanding of the law.