We develop a package of numerical simulations implemented in MATLAB to solve complex many-body quantum systems. We focus on widely used examples that include the calculation of the magnetization dynamics for the closed and open Ising model, dynamical quantum phase transition in cavity QED arrays, Markovian dynamics for interacting two-level systems, and the non-Markovian dynamics of the pure-dephasing

A numerical model implemented in the open-source finite-element method FreeFEM program is presented, with the aim of introducing students to the calculation of AC losses in superconductors. With this simple approach, students can learn about the critical state model used to describe the macroscopic electromagnetic behavior of superconductors and the importance of different factors influencing the AC

We calculate the electrostatic potential and electric field of a uniformly charged disk everywhere in space. This electrostatic problem was solved long ago, and its gravitational analogue was solved even earlier. However, it seems that physics students are not aware of the solution, because it is not presented in textbooks. The purpose of the present article is to fill this gap in the pedagogical literature

Operators play an important role in the formalism of quantum mechanics. By means of the technique of integration within an ordered product of operators and Dirac notation, we analytically prove that a new kind of asymmetric two-mode projection integral operator and another asymmetric two-mode Fock state projection summation operator are equal to the same Hermitian operator. For such Hermitian operator

The x-ray diffraction (XRD) technique has become an irreplaceable tool in the study of crystalline solids and is taught as part of the content of undergraduate solid state physics and chemistry courses. The XRD by crystalline materials is a scientific discovery predicted by von Laue, experimentally verified by Friedrich and Knipping and further developed by Bragg and Bragg, father and son, all in 1912

Teaching introductory quantum mechanics is challenging, especially to students who have limited knowledge of mathematics. The challenges and approaches are evidenced in a wide range of literature. Our goal is to develop an active engagement module for introductory quantum mechanics that would cover the most important concepts: indeterminism, incompatibility, the difference between superposition and

In this paper, we consider a small ball placed in a liquid-filled transparent cylinder sealed at both ends. When the tube is inclined and spun about the vertical axis like a centrifuge, a heavy ball (more dense than the liquid) will accelerate to either the top or bottom end of the tube depending on its release position while a light ball (less dense than the liquid) will instead tend towards an equilibrium

This paper offers a critical examination of the classical method of introducing the elecrtromagnetic potentials in a nonrelativistic context. It applies the nonretarded Helmholtz theorem to show that the usual method is ambiguous, with that ambiguity being reflected in the gauge transformation equations. This ambiguity can be removed by carefully invoking the Helmholtz decomposition theorem, but the

The geodynamo usually appears as a somewhat intimidating subject. Its understanding seems to require knowledge of the intricate theory of magnetohydrodynamics. The solution of the corresponding equations can only be achieved numerically. It seems to be a subject for the specialist. We show that one can understand the basics of the functioning of the geodynamo solely by using the well-known laws of

In physics, consideration of phenomena in one or two space dimensions (1 + 1 or 2 + 1 space-time dimensions) is often regarded as a useful step toward understanding them in three space dimensions (3 + 1 space-time dimensions). This article presents the basic dynamics of a neutron with a magnetic moment in the presence of certain electromagnetic fields in 2 + 1 space-time dimensions by using a polar

We point out that a typical two-electron distribution function in atoms and molecules, often called the intracule, has a particular sensitivity to the electron–electron repulsion, which leads to the so-called Coulomb correlation. The difference between the intracule densities is computed using the correlated and uncorrelated wave functions, and has been given the name Coulomb hole. The study of intracule

In 1983 the speed of light was set as an exact quantity. It is now one of the fundamental constants in physics, with the meter being directly related to this speed and the definition of the second. As such, experiments that calculate the speed of light with high precision are important in university undergraduate laboratories. In the experiment discussed here, a method is described that allows the

The Mori–Zwanzig projection operator formalism is one of the central tools of nonequilibrium statistical mechanics, allowing one to derive macroscopic equations of motion from the microscopic dynamics through a systematic coarse-graining procedure. It is important as a method in the physical sciences and gives many insights into the general structure of nonequilibrium transport equations and the general

Chirality, or handedness, is a topic that is common in biology and chemistry, yet is rarely discussed in physics courses. We provide a way of introducing the topic in classical physics, and demonstrate the merits of its inclusion—such as a simple way to visually introduce the concept of symmetries in physical law—along with giving some simple proofs using only basic matrix operations, thereby avoiding

We analyse the velocity of a particle propelled by a constant power and subjected to a dissipative force. We consider three cases of dissipative processes: dry friction and viscous drag in the regime of linear and quadratic dependence on velocity. We derive the equation of motion and then, based on a rational and graphical analysis, determine the qualitative shape of the velocity curve as a function

This paper presents a series of experiments that aim to aid with the understanding of some of the fundamental concepts pertaining to the resonance of cantilevers. The goal is to provide an accessible platform for exploring how the cantilever, one of the fundamental components in non-contact scanning probe/atomic force microscopy, is utilized in measuring forces. A straightforward experiment that utilizes

We illustrate how the dynamics of a system of moving mechanical objects, in interaction with the electric currents in electromagnets, can be modelled by use of the action principle. After defining the model by a Lagrange function, an internally consistent set of dynamical equations can be derived in a straightforward manner. We also illustrate the philosophy of effective descriptions , where certain

The full details of a newly developed versatile, portable and low-cost magneto-optical device for the accurate determination of Verdet constant over the wavelength range of 400–480 nm in undergraduate physics and chemistry labs are presented. The elevated sensitivity of the device is achieved by using a semiconducting photodiode detector coupled with an electronic circuit board, which is configured

In this paper, we examine the implementation of a digital learning environment—namely, the physics software, Algodoo —which is less-constrained in its design than the digital learning environments typically used in physics education. Through an analysis of a case study, we explore a teaching arrangement wherein physics teachers responsively guide small groups of students as they use less-constrained

The impact of a ball with a flexible beam is analysed both experimentally and theoretically. The impact generates a bending wave that may or may not return to the impact point before the ball bounces off the beam. If the ball rebounds before the bending wave returns, then the rebound speed of the ball is determined by localised bending of the beam in the vicinity of the impact point. For a short duration

This paper presents a simple setup to investigate the continuous gas dilution process. The setup consists of vessels and an oxygen sensor. Two experiments have been conducted. First, by flowing nitrogen (N 2 ) into a vessel containing ambient air, the oxygen (O 2 ) concentration depletes following an exponential decay. The decay constant depends on the vessel’s volume and the flow rate. Second, in

The mathematical analogy of classical and matter waves can help to teach the elusive subject of tunneling times in undergraduate physics courses. The tunneling of mechanical energy through a taut string is revisited in this paper in order to study tunneling times for this classical system. General properties of the group delay, the dwell time and the interference time are described. Moreover, we explain

The item response curve (IRC) shows the proportion of item responses plotted against total scores achieved in an assessment instrument. It has acquired widespread employment in the field of physics education to analyze tests on physics content knowledge. In the current study, the IRC was lengthened to investigate attitude items of Colorado Learning Attitudes about Science Survey (CLASS) with a five-point

We propose an alternative way to introduce supersymmetric quantum mechanics to physics students using a point canonical coordinate transformation. We provide two point canonical transformations that allow us to obtain new supersymmetric partner Hamiltonians with iso-spectral and non-iso-spectral energy eigenvalues starting from a known solvable Hamiltonian. We illustrate this new approach by applying

In his first 1905 article on special relativity, clock synchronization by means of light rays was used by Einstein to derive the Lorentz transformations ( Ann. Phys., Lpz. 322 891–921). However, the same goal can be achieved by using bodies in free motion to synchronize clocks. To this end, one has to accept the principle of relativity, the law of inertia and the existence of a limit value for the

It is well known, from Newtonian physics, that apparent forces appear when the motion of masses is described using a non-inertial frame of reference. The generalized potential of such forces is rigorously analyzed, focusing on their mathematical aspects.

Our goal is to present a derivation of Lorentz’s transformations from gedanken experiments with electromagnetic forces without having to assume Einstein’s postulate of the constancy of the speed of light. Instead, we propose a different second postulate, namely that electric charge is independent of the state of motion of the system, and we restrict the first postulate in order to state the relativity

In this paper, we describe the instrumentation and theory of the parametric resonance generation of two coupled spring oscillators. We demonstrate the generation of the parametric resonance of coupled oscillators driven by a periodic force combined with periodic noise. We experimentally and theoretically investigated the parametric oscillations and resonances of coupled spring oscillators. Thus, a

In this paper, we show that introductory physics students may initially conceptualise Cartesian coordinate systems as being fixed in a standard orientation. Giving consideration to the role that experiences of variation play in learning, we also present an example of how this learning challenge can be effectively addressed. Using a fine-grained analytical description, we show how students can quickly

Computational modelling is not only an important element in scientific research, it also has a rich history in physics education and its application in the classroom has changed greatly over the last few decades. To explain why computational modelling is used in physics education, we discuss its use in teaching and the benefits to the teaching and learning process. Secondly, historical developments

Considerations of optical damage induced by thermal effects were initiated by an apparent paradox: small and large diameter lenses with the same f -number produce an image of the Sun with the same irradiance. Thus, if one assumes that a damage threshold is given by irradiance (or intensity, as it is specified by most laser optics companies, in relation to continuous-wave lasers), all digital cameras

We make some crucial remarks about the recent presentation by Cross (2019 Eur. J. Phys. 40 065002) for pedagogical purposes considering his results. The possible deformations will influence significantly the role of (high) spots played in an abrupt stop of Euler’s disk as claimed by Cross. We thought Cross should consider the crucial role of the deformations (in a spinning disk as well as the surface

In this work we present Langevin’s original method to study the Brownian motion of a charged particle across a constant magnetic field. Langevin’s strategy allows one to transform the Langevin equation into a deterministic one associated with mean square displacement (MSD), whose solution is explicitly studied in this work. We calculate the MSD for the z -axis, along which the magnetic field is allowed

A little-known property of free-fall motion is the elliptic locus of the maximum heights attained by coplanar projectiles launched from a single point in different directions with the same initial speed. Another, less-known, and perhaps somewhat surprising, property of this family consists in the said ellipse being also the geometric locus of the critical points of the projectiles’ distances from the

In the elementary theory of electrical circuits, the series connection of n identical resistors has an equivalent resistance n 2 greater than the equivalent resistance of their parallel connection. More briefly, the series/parallel ratio is n 2 . We show that if the resistors are not all identical, the series/parallel ratio is greater than n 2 and can never be less than n 2 . This little-known minimum

The traditional pedagogical paradigm in physics is based on a deductive approach. However, with the recent advances in information technology, we are facing a dramatic increase in the amount of readily available information. We propose that incorporating inductive elements in teaching—by infusing qualitative methods—will better prepare us to deal with the new information landscape and the realities

A derivation of the space-time expression of the Pauli–Jordan function (PJF) was shown and a more compact result was obtained. Compared to other derivations, the approach utilizes more physical image, such as a Lorentz invariance of the PJF. Furthermore, the approach does not need sophisticated mathematical techniques and is therefore more suitable for the beginner of quantum field theory, such as

In every beginner electricity and magnetism course, a good teacher usually insists that if one has an infinite charged plate or an infinity charged wire, one must first calculate the electric field to find the potential difference between two space points by making a linear integral of the field. One does it this way because Coulomb’s potential, i.e. the expression which allows for a direct calculation

The volume charge density for a conducting ellipsoid is expressed in simple geometrical terms, and then used to obtain the known surface charge density as well as the uniform charge per length along any principal axis. Corresponding results are presented for conducting hyperellipsoids in any number of spatial dimensions. The presentation is at a level suitable for use in graduate courses on electrostatics

We study the electrostatic responses (i.e. retardation effects due to the propagation of electromagnetic waves are ignored) of a linear homogeneous and anisotropic (LHA) dielectric film to an arbitrary external electrostatic potential. A set of algebraic equations has been established to calculate the polarisation charges induced in the film. In our derivation, the idea is exploited that a physical

In this paper we consider the properties of the transient process that always precedes (or completes) the onset (cessation) of steady fluid flow in a uniformly rotating vertical cylinder. First, we give an elementary (without using the Navier–Stokes equations) and detailed derivation of the equation describing the space and time distribution of angular speeds in a rotating viscous fluid. This expression

The shape of a liquid surface in a rotating frame depends on the angular velocity. In this experiment, a fluid in a narrow rectangular container is placed on a rotating table. A smartphone fixed to the rotating frame records the fluid surface with its camera and, with a built-in gyroscope, the angular velocity. As the table rotates the surface evolves and develops a parabolic shape. Using video analysis

A simple and easy to implement method for improving the convergence of a power series is presented. We observe that the most obvious or analytically convenient point about which to make a series expansion is not always the most computationally efficient. Series convergence can be dramatically improved by choosing the center of the series expansion to be at or near the average value at which the series