Table of contents

An introduction to the Zwanzig–Mori–Götze–Wölfle memory function formalism (or generalized Drude formalism) is presented. This formalism is used extensively in analyzing the experimentally obtained optical conductivity of strongly correlated systems such as cuprates and iron-based superconductors. For a broader perspective both the generalized Langevin equation approach and the projection operator approach for the memory function formalism are given. The Götze–Wölfle perturbative expansion of memory function is presented and its application to the computation of the dynamical conductivity of metals is also reviewed. This digest of the formalism contains all the mathematical details for pedagogical purposes.

The International Physicists' Tournament (IPT) 2019 dealt with 17 challenging problems. In this article, we present experimental as well as theoretical approaches as examples of one of these tasks. A ball placed on a hard and flat surface can oscillate when being hit by a jet of water from above. To explain the oscillations, a theoretical model is introduced and its predictions are compared to experimental measurements. Furthermore, the structure of the IPT itself is characterized briefly in this article, as well as the idea of a new and innovative seminar, which was set up at Erlangen University to prepare and assist students in taking part in the IPT. Furthermore, the educational relevance of physics tournaments such as IPT for physics education at university is discussed in detail.

Several natural phenomena are ruled by the same first-order differential equation. Curiously, the Friedmann equation of relativistic cosmology is of the same form, allowing for useful analogies to be built. Several applications of this equation to phenomena occurring in nature are presented.

Dissipation is one of the important factors determining the motion of a gyroscope or heavy symmetric top in real life. In this work, for such motions, friction at the touchpoint with horizontal surface and drag are approximated with simple models, and they are considered as torque in Euler equations. Using this model, we study the motion of a gyroscope and a heavy symmetric top numerically. For the gyroscope, the effect of dissipation is studied while it is spinning without nutation and precession. For the heavy symmetric top, we consider two different types of motion, obtained experimentally in an earlier work in the literature. The numerical results show that results similar to those achieved experimentally, with the exception of the dissipative change in inclination angle, can be obtained numerically via this simple model, by utilizing Euler equations.

The effects of various types of damping on the period of the nonlinear pendulum are compared with experiment for oscillations of amplitude up to ∼180°. The factors affecting a comparison of experimental results obtainable with modern MEMs gyro/accelerometers with theory, namely a precise knowledge of the pendulum fundamental frequency ω₀, the instrument calibration and the definition of the amplitude of the damped oscillation are presented together with the damping theory.

Measurements are presented of the drag force on several different balls falling vertically through water in a fish tank, at speeds up to 4.5 m s⁻¹. Experiments of this type are usually undertaken in a student laboratory by measuring the terminal velocity of an object falling either through air or a viscous liquid. At speeds slightly higher than the terminal velocity, additional effects can be observed visually in water, including flow separation, turbulent flow and the formation of air cavities. Changes in the drag coefficient were observed due to all three effects, although different changes were observed with different balls, depending on the Reynolds number. Cavity formation was found to depend strongly on the roughness of the ball surface as well as the incident ball speed. Results are presented at Reynolds numbers from about 20 000 up to about 250 000.

Harmonic motion is a significant topic in undergraduate lectures for university physics students. A typical example, presented in lectures around the globe, is the motion of a mass at the end of a spring. It should be emphasized that the harmonic motion in the aforementioned case is just an approximation for small displacements. However, the value of this example for teaching purposes is undeniable, if used properly. Its significance is connected to the adoption of an analytical way of thinking by the students that can help them to not only memorize isolated fragments but to really understand the topic in depth. As a result, the students should be able to apply a similar analysis to new and unsolved problems and explore unknown physical systems using basic theoretical tools. To move in the right direction, a new example is presented in this paper: the motion of a rigid sphere on an elastic half space. This phenomenon is extremely complicated. However, in this paper it is shown that for a specific range of values, this motion can be approximated to harmonic motion. This fact will significantly help students to learn how to resolve unknown problems and tasks using their knowledge. In addition, the analysis provided by this paper can be used as an alternative and simpler solution for data processing in nanoindentation experiments. As a result, it can be used as a valuable tool not only for undergraduate physics students but also for young researchers in the fields of nanotechnology and materials science.

A thermometer is a temperature proxy, usually with a fixed number of parameters. When a nearly ideal gas or a similarly simple system is not available, the determination of a temperature scale is difficult, and has been subject to debate. Entropy uniqueness, due to Clausius, provides a fundamental theoretical criterion to answer this question. By measuring the heat flow dQ around a grid in p–V space, with M elemental areas, one can test a proposed temperature scale T_n by computing $\sum {\left[\oint \left(\mathrm{d}Q/{T}_{n}\right)\right]}^{2}$, the rms deviation, from the desired value of zero. This approach also can be used to determine which of a group of n thermometers has the most accurate temperature scale T_n. As background we discuss Clausius's development of the mathematics and the concept of entropy. An appendix shows how energy conservation permits one to eliminate the zigs and zags that appear in the Clausius construction for a general Carnot cycle.

A simple experiment based on thermal imaging is devised to observe and measure, along a cooling fin in contact with a hot spot, the temperature profile determined by conductive and convective heat transport. By using a simple theoretical approach, the cooling fin equation is justified and applied to describe, in an educational framework and perspective, the experimental results. Both the experimental setup and the theoretical model discussed in this work are suited for undergraduate physics students.

Nonrelativistic classical mechanics allows no fundamental transition between low-velocity and high-velocity forms of behavior, nor between low-temperature and high-temperature forms. In contrast, classical electrodynamics, which is a relativistic theory, allows fundamental transitions in velocity. Furthermore, the inclusion within the classical theory of Lorentz-invariant classical zero-point radiation allows classical electrodynamics to distinguish high-temperature and low-temperature forms of behavior. Because electromagnetism is a relativistic theory, it may provide a thermal radiation bath which gives rise to phenomena in Brownian motion which are not included in a model with a thermal bath based upon nonrelativistic particles. Here we explore the Brownian motion of a classical electric-dipole particle in random classical radiation, making use of the calculations of Einstein and Hopf. The Brownian motion as a function of temperature is analyzed in terms of the mean-square velocity and the diffusion constant for four different classical radiation spectra: the Rayleigh–Jeans spectrum, the Planck spectrum without zero-point radiation, the zero-point radiation spectrum, and the Planck spectrum including zero-point radiation. We illustrate how the inclusion of classical electromagnetic zero-point radiation alters Brownian motion behavior between high-temperature and low-temperature forms. For the Planck spectrum with zero-point radiation, the high-temperature Brownian motion agrees with some aspects found from nonrelativistic mechanics, while the low-temperature behavior includes some aspects analogous to superfluid behavior. At sufficiently low temperatures, the Brownian particle has an increasing mean-square velocity and more rapid diffusion with decreasing temperature due to the increasing dominance of the classical electromagnetic zero-point radiation.

A solar thermal experiment for the thermal performance evaluation of parabolic dish solar cookers suitable for developing countries is presented. Three different solar cooking pots are compared experimentally using three different cooking fluids, namely water, sunflower oil and olive oil. The heating experiments are carried using three identical parabolic dish solar cookers. The mass of the fluid load used in the experimental tests is 2.5 kg, on each test day. The pots that are experimentally tested under Mahikeng, South Africa, conditions are a black stainless steel pot, an aluminium silver pot and a copper pot. The stainless steel black pot shows higher temperatures as compared to the other pots. Higher energy efficiency values are also obtained with the parabolic dish solar cooker using the black stainless steel pot. The absorbing colour of the cooking pot has a more significant effect on the cooking performance than thermal conductivity of the pot. The best heat transfer fluid which obtains higher temperatures and higher cooking efficiencies is sunflower oil. The aluminium silver pot shows the lowest energy efficiencies in almost of all of the experiments. For Mahikeng conditions, the greatest cooking potential in terms of the speed of cooking is shown by the black pot using sunflower oil as the heat transfer fluid. The thermal performance in the cooking experiments is affected by ambient conditions and manual solar tracking. To shorten cooking times, a larger parabolic dish solar cooker can be used.

Columns of water in inverted containers, such as upside-down glasses of water, are classic phenomena that demonstrate the effects of atmospheric pressure. This paper presents an analytical method to explore the role of liquid bridges and the associated capillary forces in the equilibrium condition and the criticality of water columns held in inverted containers by a cover sheet. This method has been validated by comparing with experimental data and previous methods. Furthermore, we found that the cover sheet was not separated from the water column, but instead fell together with it, during the drop. The analytical calculation shows that the water is prone to flow out with the glass half full of water, and the cover sheet is prone to attach to a wider or a shorter glass. Furthermore, we found the conditions for pressure balance of the cover sheet and of the water column are different. The cover sheet does not fall down due to differences in pressure between the upper and lower surfaces. The static balance of the water column in the glass is dependent on balancing the additional pressure due to the surface tension of the water.

We analyze the propagation properties of 2D electromagnetic beams related to the symmetry of the electric field on their waist. We find conditions under which they can be adequately represented by the propagation of a single planewave component; we also exhibit examples where this is not so, and, thus, care has to be taken when using refraction and reflection laws to determine the refraction and reflection angles of a beam.

The teaching of basic circuits in early undergraduate physics courses is widespread and, to some extent, quite uniform. In most textbooks, the introduction of resistors, capacitors and inductors is followed by their assemblage into larger systems, together with mathematical calculations of charges and currents as functions of time. In this process, spatial features of circuits tend to be omitted. Here, we argue that this kind of omission is not 'natural' but, rather, should be justified in terms of internal time-scales operating in the system, in the framework of the continuity equation. When space is brought back to the discussion of circuits, a number of important physical features, absent in most textbooks, emerge. Among them, one has a spatial uniformity of currents, the existence of charges coating metallic surfaces, and the presence of charge distributions inside wires at the endings of resistors and coils. We believe that the discussion of these qualitative issues with students could foster a more mature and unified relationship with electromagnetism.

We describe gauge symmetry at the elementary level aiming at undergraduate and first-year graduate theoretical physics students. The formal definition of gauge invariance as a local symmetry of the action is given and further elaborated and interpreted in detail in two physical models, namely, a mechanical system and the prototypical gauge field theory of electrodynamics. The first model, consisting of two interacting particles with gauge symmetry, sets the background and paves the way for the canonical analysis of the electromagnetic field as a constrained system. The Dirac–Bergmann algorithm and the necessity of Dirac brackets for the canonical quantization of constrained systems are discussed within the two models context. The whole analysis is written in a didactical way trying to guide and motivate undergraduate students along an introduction to gauge symmetry. After going through this present brief introduction, the reader should be well prepared to boldly continue their journey through more advanced standard textbooks on the subject.

We developed a microdisplay-based microscope projection photolithography (MDMPP) technique in which a liquid crystal (LC) microdisplay is used as a reconfigurable photomask for a microscope projector. The LC microdisplay provides a significant advantage in terms of cost and speed since patterns can be generated through software instead of redesigning and fabricating glass photomasks. The constructed MDMPP system could produce line patterns as narrow as 2.4 µm, smaller than that specified by the diffraction limit, with the aid of a 4× objective lens. The achievement of a linewidth smaller than the theoretical limit may be ascribed to a combination of overexposure and the underetching effect, in addition to the good optical performance of the system. In a diffraction experiment performed with fabricated slits, the application of the MDMPP technique helped provide various patterns of the slits, demonstrating the potential usefulness of the MDMPP system in undergraduate optics courses. We expect that MDMPP can contribute to the field of physics education and various areas of research, such as chemistry and biology, in the future.

The efficiency of students' educational activity in a physical laboratory increases if students independently assemble experimental installations and use them to study physical phenomena. This paper describes a simple installation for the manufacture of a sinusoidal grating by the holographic method. Such a grating is used in many demonstration and laboratory experiments. It is used, for example, in a simple device for making photos of optical spectra on a smartphone, and this device is also described in this paper. First, students study the elementary theory of a holographic diffraction grating. They use this theory to calculate the parameters of the installation that allows them to obtain a grating with the required period and use an experiment to verify the validity of the theory. Then they calculate the parameters of the spectroscope on the basis of the studied theory and again make sure that the theory is valid. Thus, it is possible to organize a learning process in which the theory is fully confirmed by the experiment.

Many students majoring in science may not have a chance to directly observe spherical aberration or understand its mechanism during their study of optics. In this paper, the author reports a method of directly observing spherical aberration under a microscope by comparing focal spot images of spherical mirrors and parabolic mirrors made of celluloid concave micromirrors, fabricated by Suzuki's universal microprinting method, invented in Japan in 1930. The application of concave micromirrors to the optical-axis alignment of a microscope light source is also explained. Furthermore, the application of concave micromirrors to a substrate of micro-watch glass, used in microorganism microscopy, is also explained.

This paper presents an educational concept for promoting quantum teaching and learning via an educational structured quantum optical experiment. The experiment is designed to demonstrate the striking differences between classical physics and quantum physics, for example quantum interference of unbreakable photons (the ability of probabilities to interfere due to a phase sensitive superposition of states) and quantum nonlocality (there is no way to locate photonic states without a fundamental loss of information about the characteristics and a complete change of the state). For this proposal, we developed an experimental setup straightforward enough to be used in advanced physics courses even in secondary school student labs. To explain, or in a more quantum-semantic way, to interpret the experimental results quantitatively, we provide an appropriately rigorous quantum optical theory. Our model combines Laplace statistics (to access the statistical behaviour of photon counting) and basic vector calculus to calculate probabilities from the phase sensitivity of probability amplitudes. This article aims to contribute to further discussion and empirical research into novel teaching strategies for a more deeply conceptual approach to quantum theory.

The double slit experiment was the first demonstrative proof of the wave nature of light. It was expounded by the English physician-physicist Thomas Young in 1801 and it soon helped lay to rest the then raging Newton–Huygens debate on whether light consisted of a fast-moving stream of particles or a train of progressive waves in the ether medium. In the experiment, light is made to pass through two very narrow slits spaced closely apart. A screen placed on the other side captures a pattern of alternating bright and dark bands called fringes which are formed as a result of the phenomenon of interference. In prior work by the same author, it was shown that the conventional analysis of Young's experiment that is used in many introductory physics textbooks, suffers from a number of limitations in regards to its ability to accurately predict the positions of these fringes on the distant screen. This was owing to the adoption of some needless and paradoxical assumptions to help simplify the geometry of the slit barrier-screen arrangement. In the new analysis however, all such approximations were discarded and a hyperbola theorem was forwarded which was then suitably applied to determine the exact fringe positions on screens of varied shapes (linear, semi-circular, semi-elliptical). This paper further builds on that work by laying down the mathematical framework necessary for counting fringes and then comparing their distributions on differently shaped screens, using MATLAB software package for numerical–graphical simulation. In addition, a pair of equivalent laws of proportionality are predicted that govern the distribution of fringes independent of the shape of the detection screen employed.

The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice on which calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger's equation. The theoretical and computational framework presented here introduces undergraduates to the path integral formulations of quantum mechanics in real time and the partition function in statistical mechanics in imaginary time. The example of the anharmonic oscillator helps to build an intuition about the MCMC method of evaluating the partition function, which can then be used to solve other problems in physics and beyond.

The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic ($\frac{1}{2}k{x}^{2}$) and quartic (λx⁴) terms is embedded within an infinite square well potential of appropriate width, 'a' and its sine eigen functions are used as basis functions 'N' for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters 'a' and 'N' are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, 'λ'. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.

The delayed-choice quantum eraser has long been a subject of controversy, and has been looked at as being incomprehensible to having retro-causal effect in time. Here the delayed-choice quantum eraser is theoretically analyzed using standard quantum mechanics. Employing a Mach–Zehnder interferometer, instead of a conventional two-slit interference, brings in surprising clarity. Some common mistakes in interpreting the experiment are pointed out. It is demonstrated that in the delayed mode there is no which-way information present after the particle is registered on the screen or the final detectors, contrary to popular belief. However, it is shown that another kind of path information is present even after the particle is registered in the final detectors. The registered particle can be used to predict the results of certain yet to be made measurements on the which-way detector. This novel correlation can be tested in a careful experiment. It is consequently argued that there is no big mystery in the experiment, and no retro-causal effect whatsoever.

We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity.

In this paper, an algebraic solution for the bound states of the relativistic hydrogen atom is presented. The method discussed here adds an operator associated with the phase of the energy eigenstates to the set of variables of the problem. In terms of this set, appropriate ladder operators are constructed in order to express the full solution of the Dirac hydrogen equation. These ladder operators are used to form the Lie algebra of a su(1, 1) group, in the same way as that applied to angular momentum algebra and the ladder operators L_±. The elements of the vector space associated with the representation of this algebra are related to a generalization of the Laguerre polynomials of the non integer index, also known as Sonine polynomials. In addition, we find that the eigenvalues of the operator constructed with the sum of the square of the three su(1, 1) generators gives precisely the relativistic energy spectrum of the hydrogen atom, including its angular momentum dependency.

We have designed and fabricated an inexpensive AC susceptometer which works around room temperature. The susceptometer uses the principle of a mutual inductance bridge with a null detection algorithm. The bridge was balanced by two digitally controlled stepper motors coupled with two highly linear potentiometers. The susceptometer was tested with double-perovskite structured samples with transition temperatures close to room temperature. The apparatus would be very useful for low-budget research laboratories which deal with magnetic experiments.

We consider the Thomas–Wigner rotation of coordinate systems under successive Lorentz transformations of inertial reference frames, and disclose its physical mechanism on the basis of the relativistic contraction of moving scale, and the relativity of the simultaneity of events for different inertial observers. This result allows us to better understand the physical meaning of the Thomas precession, and to indicate some overlooked aspects of the physical interpretation of this effect, as related to two specific examples: the circular motion of a classical electron around a heavy nucleus, and the motion of a classical electron along an open path, where its initial velocity and acceleration are mutually orthogonal to each other.

Grading can shape students' learning and encourage use of effective problem solving practices. Teaching assistants (TAs) are often responsible for grading student solutions and providing feedback, thus, their perceptions of grading may impact grading practices in the physics classroom. Understanding TAs' perceptions of grading is instrumental for curriculum developers as well as professional development leaders interested in improving grading practices. In order to identify TAs' perceptions of grading, we used a data collection tool designed to elicit TAs' considerations when making grading decisions as well as elicit possible conflicts between their stated goals and actual grading practices. The tool was designed to explicate TAs' attitudes towards research-based grading practices that promote effective problem solving approaches. TAs were first asked to state their goals for grading in general. Then, TAs graded student solutions in a simulated setting while explicating and discussing their underlying considerations. The data collection tool was administered at the beginning of TAs' first postgraduate teaching appointment and again after one semester of teaching experience. We found that almost all of the TAs stated that the purpose of grading was formative, i.e. grading should encourage students to learn from their mistakes as well as inform the instructor of common student difficulties. However, when making grading decisions in a simulated setting, the majority of TAs' grading considerations focused on correctness and they did not assign grades in a way that encourages use of effective problem solving approaches. TAs' perceptions of grading did not change significantly during one semester of teaching experience.

We describe in detail a high-performance project devised for outstanding undergraduate students with appropriate abilities in physical reasoning (rather than with a good standard preparation), centered around the well-known historical case of Newton's theory of light and colours. The different action lines along which the project is developed are aimed at involving the students in: (1) thinking as Newton did, by building step by step all of his knowledge and reasoning; (2) working as Newton did, by performing the whole series of his original experiments with prisms; (3) deducing, as Newton did, the nature of light and colours; (4) presenting the results of their activity (including physics demonstrations) to the general public, in order to test their ability to communicate what they learned and discovered (including video realization published on YouTube [1]). Such teaching aims are complemented by the purpose of realizing a historically informed activity, given the potential key role of the history of physics in promoting science at a deeper level, particularly when no particular training in mathematics or advanced education is required. A testament to the success of this work can be seen in terms of the students' enthusiasm when they demonstrated their work at public events, and the ready involvement of people without professional or specialised knowledge in physics attending the events.

In their famous 1939 paper on the physics of nuclear fission, Niels Bohr and John Wheeler presented an eye-catching graph of the fission barrier energy for various nuclides. In this paper I develop a simplified approach to developing this graph which is both sensibly true to their analysis and appropriate for a student audience.

Composites based on candle wax and pencil graphite were fabricated with different concentrations of graphite in the range 0% w/w to 70% w/w. These composites were fabricated ('ironed') into a sandwich structure with kitchen aluminum foil serving as electrodes. The composites' electrical resistance was measured and it was found that the composite with 15% w/w of graphite was the best choice for further investigation according to the percolation theory of electrical conductivity thresholds. The wax–graphite composites were subjected to impact forces from a basketball, measured by the school's PASCO equipment. A calibration curve of the specific electrical resistance versus the force of impact was constructed. Following the standards of the International Basketball Federation (FIBA), the quality of the basketball was evaluated in terms of the bounce height of a free-falling ball and the force acting on the ball during the impact with the floor. The wax–graphite composite with a threshold concentration of 15% w/w graphite proved to be a sensitive sensor for measuring the impact force, even when small forces were under investigation. The project as presented here could be used as a laboratory topic for advanced level high school physics or undergraduate lab work in materials science or applied physics.

In this article, we present the development and treatment of an inverse problem applied to data from the Felix Baumgartner stratospheric jump. This jump is well documented with a lot of data. Therefore, it makes a particularly well-suited example for teaching in a numerical techniques laboratory. The major aim of the article is to give guidelines in order to construct a simple, but not simplistic, inverse problem with real data for junior undergraduate students. Students should master classical mechanics and have some skills in numerical modelling. We use the programming language Python and various libraries in order to build a model and solve the entire problem. This programming language is increasingly used and understood by students, which allows them to focus on the physical and numerical aspects of the involved problem. The fairly new strategy presented in this article is an attempt to estimate the angle of attack from acceleration measurements, and to give an uncertainty estimation of Baumgartner's free-fall speed.

This comment examines an article that develops a counterclockwise reversible Stirling cycle whose coefficient of performance equals that of a Carnot refrigerator. Unfortunately, among other issues, the proposed cycle is never in thermodynamic equilibrium and thus cannot match an idealized reversible Carnot.