Table of contents

We examine the Stark effect (the second-order shifts in the energy spectrum due to an external constant force) for two one-dimensional model quantum mechanical systems described by linear potentials, the so-called quantum bouncer (defined by V(z) = Fz for z > 0 and V(z) = ∞ for z < 0) and the symmetric linear potential (given by V(z) = F|z|). We show how straightforward use of the most obvious properties of the Airy function solutions and simple Taylor expansions gives closed form results for the Stark shifts in both systems. These exact results are then compared to other approximation techniques, such as perturbation theory and WKB methods. These expressions add to the small number of closed-form descriptions available for the Stark effect in model quantum mechanical systems.

Berry's phase, although well known since 1984, has received little attention among textbook authors of solid state physics. We attempt to address this lack by showing how the presence of the Berry's phase significantly changes a standard concept (effective mass) found in most solid state texts. Specifically, we show that the presence of a non-zero Berry curvature in Bloch systems makes the traditional concept of an inverse effective mass tensor M⁻¹ problematic, since a routine application of Newton's second law leads to a circular definition. As a consequence, the related concept of cyclotron effective mass m* also requires modification. It is shown that m* is magnetic-field dependent in non-inversion symmetric systems. This has important ramifications for cyclotron resonance experiments, since such experiments yield m* and thereby purportedly give the components of M⁻¹. This work represents a 'case study' in how Berry's phase effects can modify 'standard' solid-state topics in ways that students and instructors may find surprising.

This paper describes an educational analysis of a first year physics experiment on standing waves for engineering students. The educational analysis is based on the ACELL (Advancing Chemistry by Enhancing Learning in the Laboratory) approach which includes a statement of educational objectives and an analysis of student learning experiences. The experiment is likely to be found in many physics departments, hence is appropriate to illustrate the ACELL approach in physics. The concepts associated with standing waves are difficult; however, they are underpinned by mathematical formulation which lend themselves to be visualized in experiments. The challenge is to strike a balance between these two for the particular student cohort. In this study, this balance is achieved by using simple equipment and providing appropriate scaffolds for students to associate abstract concepts with concrete visuals. In essence the experiment is designed to adequately manage cognitive resources. Students work in pairs and are questioned and assisted by demonstrators and academic staff during a 2 h practical class. Students were surveyed using the ACELL instrument. Analysis of the data showed that by completing the practical students felt that their understanding of physics had increased. Furthermore, students could see the relevance of this experiment to their engineering studies and that it provided them with an opportunity to take responsibility for their own learning. Overall they had a positive learning experience. In short there is a lot of dividend from a small outlay of resources.

A detailed derivation of analytic solutions is presented for overlap, kinetic, nuclear attraction and electron repulsion integrals involving Cartesian Gaussian-type orbitals. It is demonstrated how s-type orbitals can be used to evaluate integrals with higher angular momentum via the properties of Hermite polynomials and differentiation with respect to non-integration variables.

The transmission through a stack of identical slabs that are separated by gaps with random widths is usually treated by calculating the average of the logarithm of the transmission probability. We show how to calculate the average of the transmission probability itself with the aid of a recurrence relation and derive analytical upper and lower bounds. The upper bound, when used as an approximation for the transmission probability, is unreasonably good and we conjecture that it is asymptotically exact.

We discuss the categorization of 20 quantum mechanics problems by physics professors and undergraduate students from two honours-level quantum mechanics courses. Professors and students were asked to categorize the problems based upon similarity of solution. We also had individual discussions with professors who categorized the problems. Faculty members' categorizations were overall rated higher than those of students by three faculty members who evaluated all of the categorizations. The categories created by faculty members were more diverse compared to the categories they created for a set of introductory mechanics problems. Some faculty members noted that the categorization of introductory physics problems often involves identifying fundamental principles relevant for the problem, whereas in upper-level undergraduate quantum mechanics problems, it mainly involves identifying concepts and procedures required to solve the problem. Moreover, physics faculty members who evaluated others' categorizations expressed that the task was very challenging and they sometimes found another person's categorization to be better than their own. They also rated some concrete categories such as 'hydrogen atom' or 'simple harmonic oscillator' higher than other concrete categories such as 'infinite square well' or 'free particle'.

We discuss a quantum-mechanical model of two particles that interact by means of a harmonic potential and are confined to a one-dimensional box with impenetrable walls. We apply perturbation theory to the cases of different and equal masses and analyse the symmetry of the states in the latter case. We compare the approximate perturbation results with accurate numerical ones.

The history of Laplace's equations for spherical and cylindrical droplets and the concept of dividing surface in Gibbs' thermodynamic theory of capillary phenomena are briefly reviewed. The existing theories of surface tensions of cylindrical droplets are briefly reviewed too. For cylindrical droplets, a new method to calculate the radius and the surface tension of the surface of tension is given. This method is suitable to be used by molecular dynamics simulations.

Dispersive forces are a kind of van der Waals intermolecular force which could only be fully understood with the establishment of quantum mechanics and, in particular, of quantum electrodynamics. In this pedagogical paper, we introduce the subject in a more elementary approach, aiming at students with basic knowledge of quantum mechanics. We perform original calculations using a semiclassical method by the name of the fluctuating-dipole method, and focus primarily on the interaction of atoms with macroscopic bodies, in particular with a spherical, less usual geometry.

The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The xⁿ(a − x)ⁿ and xⁿ(a/2 − x)(a − x)ⁿ polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.

The term 'apparent depth' is commonly treated in textbooks as an issue easily understandable from the point of view of paraxial optical geometrical optics. Nevertheless, everyday life tells us that most of the time the observation of objects immersed in water is made under a great range of visual angles where the paraxial approximation is not valid. Here we developed a non-paraxial treatment to calculate the position and shape of the image of objects immersed in liquids of different refractive indices. The approach was focused on the parametric positions of the images of a single point at different viewing angles. Then we calculated how the image of an extended object is distorted. By using the Matlab software, it is possible to visualize the images for different geometrical conditions. We also include the analysis for refractive index with negative values as is the case of metamaterials.

In this paper, an experimental setup for undergraduate courses to study the damped harmonic motion of a sphere inside a fluid as a function of the medium viscosity is presented. To observe the dependence of the oscillation of the sphere on the medium viscosity, different concentrations of glycerin in water were used. The sphere is suspended on the end of a spring and its displacement is indirectly obtained using a force sensor. To describe the sphere motion, a drag force different from that given by Stokes' law is used. Our experimental results fit satisfactorily when semiempirical coefficients are introduced in the model. The frequency and relaxation time of the sphere oscillations diminish as the concentration of glycerin increases. Boundary effects due to the fluid container size are studied. We found that when the container size decreases the oscillations decay more rapidly due to a greater resistance to the motion of the sphere.

Bose–Einstein condensation (BEC) of a finite number of noninteracting bosons confined in a finite-size container is investigated by using numerical calculation. It is revealed that the characteristics of finite-size BEC are significantly different from those for the system satisfying the condition of thermodynamic limit, and some well-known conclusions about BEC described in textbooks are not valid for a finite-size Bose system. This paper may be helpful for the further understanding of the concept of BEC and thermostatistics in small systems. It is suitable for readers including undergraduate and graduate students, general physicists and so on.

The angular part of the Schrödinger equation for a central potential is brought to the one-dimensional 'Schrödinger form' where one has a kinetic energy plus potential energy terms. The resulting polar potential is seen to be a family of potentials characterized by the square of the magnetic quantum number m. It is demonstrated that this potential can be viewed as a confining potential that attempts to confine the particle to the xy-plane, with a strength that increases with increasing m. Linking the solutions of the equation to the conventional solutions of the angular equation, i.e. the associated Legendre functions, we show that the variation in the spatial distribution of the latter for different values of the orbital angular quantum number l can be viewed as being a result of 'squeezing' with different strengths by the introduced 'polar potential'.

It has already been shown that, with the simplest classical device, namely two coupled pendula, one can simulate the movement of a free quantum two-level system, as for instance a one-half spin in a static magnetic field. We now show that, with a hardly more sophisticated device, one can simulate the latter system undergoing outer perturbation. In the case of sinusoidal excitation, we present an experimental mechanical illustration of the transition of the system between its two eigenstates, known as the Rabi precession. Abstract theoretical concepts of absorption and stimulated emission of energy are thus provided with intuitive support.

In this paper, we revise the main aspects of the 'Hamiltonian analogy': the fact that optical paths are completely analogous to mechanical trajectories. We follow Schrödinger's original idea and go beyond this analogy by changing over from the Hamilton's principal function S to the wavefunction Ψ. We thus travel from classical to quantum mechanics using optics as a guide. Unfortunately, and despite its mathematical beauty and simplicity, the connection between classical and quantum mechanics through optics is nowadays hardly known and mostly ignored in university education. The present work tries to fill this gap.

Entangled states of composite quantum systems exhibit one of the most distinct and non-classical features of the quantum mechanical description of Nature, first pointed out by Schroedinger: 'Maximal knowledge of a total system does not necessarily imply maximal knowledge of all its parts'. We provide an elementary illustration of this fundamental aspect of quantum physics by considering a system of two particles in an infinite, one-dimensional square potential well. In contrast to standard introductory presentations of quantum entanglement, our present considerations do not require density matrix formalism, nor explicit use of the tensor product structure for the description of composite quantum systems.

We consider the problem of finding the optimum path of a boat traversing a straight in a current. The path of the shortest time is found using the calculus of variations with the constraint that the boat must land directly opposite to its starting point. We compare the optimal trajectory with that where the boat's local orientation is always directed to the arrival point. When analytical solutions cannot be found we use numerical methods. The level of the exposition is suitable for advanced undergraduate students, graduate students and general physicists.

The aim of this paper is to discuss in a pedagogical way the problem of the emergence of classical behaviour in certain physical systems which, in principle, are correctly described by quantum mechanics. It is stressed that the limit ℏ → 0 is not sufficient and the crucial role played by the environment must be taken into account. In particular, recalled is the old problem raised by Mott in 1929 (Proc. R. Soc. Lond. A 126 79) concerning the straight tracks observed in a cloud chamber, produced by an α-particle emitted by a source in the form of a spherical wave. The conceptual relevance of the problem for a clearer understanding of the classical limit is discussed in a historical perspective. Moreover, a simple mathematical model is proposed, where the result of Mott is obtained in a rigorous mathematical way.

This paper describes a new method of demonstrating superconductivity. A GdBaCuO high-temperature superconductor of 60 mm in diameter and 10 mm in height, a sintered NdFeB cubic magnet with an edge length of 100 mm and iron wires coated by coloured vinyl are used to visualize the magnetic field lines in the vicinity of a superconductor when it is placed in a magnetic field. The spatial distributions of iron wires make it easy to understand how the magnetism of superconductivity causes the levitation and suspension of conventional demonstrations. This paper is intended to be read especially by teachers of undergraduate courses and educators of science museums or science centres.

Even though the phase coherence is a basic principle of the Bardeen–Cooper–Schrieffer theory of superconductivity, nowhere in the literature can a direct and simple proof of it be found. Here it is shown that a minimization with respect to the phases of the probability amplitudes of the superconducting wavefunction of the theory leads naturally to phase coherence in a simple and straightforward way. The proof is given in terms of first principles and the uniqueness of the phase coherence solution is shown. This paper aims to provide a more physical insight into the origin of a concept that it is widely accepted in the above theory.

This letter deals with irreversible processes described by abrupt changes in temperature or in pressure, in which the system is placed into contact with a finite number N of heat or work reservoirs, respectively. Two previous papers have shown that when N goes to infinity the entropy generation goes to zero, i.e. the process becomes reversible. Here we look for the sequence in temperature or in pressure of such reservoirs that leads to a minimum in entropy generation and find that in both cases such a sequence is the geometric progression. This work also highlights the close relation between mathematical and physical concepts and is therefore quite interesting from an instructive standpoint.

We discuss simple examples in which renormalization is required in classical field theory. The presentation is accessible to undergraduate students with a knowledge of the basic notions of classical electromagnetism.

An ultra-capacitor in parallel with the output of a dc power supply reduces the source resistance and yields an electrical current of up to 900A through a low resistance load. The magnetic field of wires is made visible with iron filings over a large area.

This note is on Lattman's paper (2009 Eur. J. Phys.30 L41). It seems to us that the direct unification of the quantum mechanics and the gravity theory (for a black hole) by Lattman is in doubt.

In this note, we present an improved approximation to the solution of Lima (2008 Eur. J. Phys.29 1091), which decreases the maximum relative error from 0.6% to 0.084% in evaluating the exact pendulum period.

We show that the conclusion in the paper (Kholmetskii and Yarman 2008 Eur. J. Phys.29 1135–46) that there is a paradox in Poynting's theorem for a moving charged particle is wrong. Contrary to the claim in their paper, there is no need to modify the standard form of the Poynting vector for this case, and their proposed modification is wrong.

We present our reply to the criticism by Franklin (2010 Comment on 'Energy flow in a bound electromagnetic field: resolution of apparent paradoxes' Eur. J. Phys.31 L17) and show that the main body of his remarks is irrelevant.

Some errors are rectified in a recent paper by van Kampen (2008 Eur. J. Phys.29 879–83) on the force between two parallel current-carrying wires.

This reply answers the issues raised in the Comment on my paper (van Kampen 2008 Eur. J. Phys.29 879–83). The error of applying a single Lorentz transformation to a wire segment is discussed in some detail.

We revisit a classic demonstration for surface tension in soap films and introduce a more striking variation of it. The demonstration shows how the film, pulling uniformly and normally on a loose string, transforms it into a circular arc under tension. The relationship between the surface tension and the string tension is analysed and presented in a useful graphical form.

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