Table of contents

In this paper, we describe two complementary nonlinear spectroscopy methods which both allow one to achieve Doppler-free spectra of atomic gases. First, saturated absorption spectroscopy is used to investigate the structure of the 5S_1/2 → 5P_3/2 transition in rubidium. Using a slightly modified experimental setup, Doppler-free two-photon absorption spectroscopy is then performed on the 5S_1/2 → 5D_5/2 transition in rubidium, leading to accurate measurements of the hyperfine structure of the 5D_5/2 energy level. In addition, electric dipole selection rules of the two-photon transition are investigated, first by modifying the polarization of the excitation laser, and then by measuring two-photon absorption spectra when a magnetic field is applied close to the rubidium vapour. All experiments are performed with the same grating-feedback laser diode, providing an opportunity to compare different high-resolution spectroscopy methods using a single experimental setup. Such experiments may acquaint students with quantum mechanics selection rules, atomic spectra and Zeeman effect.

A vertical light ray coming from infinity is reflected by a primary parabolic mirror M₁ having focus at F₁. At a small distance from F₁ a secondary mirror M₂, symmetric with respect to the vertical axis, is placed. One would like to find the analytic equation of the mirror M₂, so that all rays from infinity, incident on M₁, could be sent back to the vertex V₁ of the primary mirror M₁. We denote the mirror M₂ with the name of zozzaroid, after P Zozzaro, who first proposed the problem. Recalling the famous legend about Dionysius ear, we find that M₂ has either an elliptic or a hyperbolic profile.

A ladder network constructed by an elementary two-terminal network consisting of a parallel resistor–inductor block in series with a parallel resistor–capacitor block sometimes is said to have a non-dispersive dissipative response. This special ladder network is created iteratively by replacing the elementary two-terminal network in place of the resistors. In this paper, it will be demonstrated that, in fact, non-dispersive dissipative response conclusion of this special ladder network is not accurate for steady-state condition. Furthermore, the voltage profile of this special ladder network exhibits a fractal form called the Lévy C curve or Lévy dragon in a certain condition. Therefore, this special ladder network may be called the Lévy ladder network. This ladder network might be interesting for physics and electrical engineering students and they may encounter when dealing with this network series and parallel combinations of impedances.

The practical applicability of a semiconductor quantum dot with spin–orbit interaction gives an impetus to study analytical solutions to one- and two-electron quantum dots with or without a magnetic field.

In this pedagogical work, we point out a subtle mistake that can be made by undergraduate or graduate students in the computation of the electrostatic energy of a system containing charges and perfect conductors if they naively use the image method. Specifically, we show that naive expressions for the electrostatic energy for these systems obtained directly from the image method are wrong by a factor of 1/2. We start our discussion with well-known examples, namely point charge–perfectly conducting wall and point charge–perfectly conducting sphere, and then proceed to the demonstration of general results, valid for conductors of arbitrary shapes.

A cubication procedure of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force, and this allows us to approximate the original nonlinear differential equation by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A, while in a Taylor expansion of the restoring force these coefficients are independent of A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain an approximate frequency–amplitude relation as a function of the complete elliptic integral of the first kind. Some conservative nonlinear oscillators are analysed to illustrate the usefulness and effectiveness of this scheme.

The Poynting vector is an invaluable tool for analysing electromagnetic problems. However, even a rigorous stress–energy tensor approach can still leave us with the question: is it best defined as E × H or as D × B? Typical electromagnetic treatments provide yet another perspective: they regard E × B as the appropriate definition, because E and B are taken to be the fundamental electromagnetic fields. The astute reader will even notice the fourth possible combination of fields, i.e. D × H. Faced with this diverse selection, we have decided to treat each possible flux vector on its merits, deriving its associated energy continuity equation but applying minimal restrictions to the allowed host media. We then discuss each form, and how it represents the response of the medium. Finally, we derive a propagation equation for each flux vector using a directional fields approach, a useful result which enables further interpretation of each flux and its interaction with the medium.

This paper presents an analysis of the thermal behaviour of objects exposed to a solar-type flux of thermal radiation. It aims to clarify certain apparent inconsistencies between theory and observation, and to give a detailed exposition of some critical points that physics textbooks usually treat in an insufficient or incorrect way. In particular, the paper examines the equilibrium temperature reached by objects exposed to solar thermal radiation and the temperature difference between their illuminated and shaded sides. These problems are studied first by analysing the simple ideal case of an isolated object, subsequently by taking into account the thermal radiation emitted by the environment, and finally by considering also the heat exchange with the surrounding air. Some examples are developed and numerical data are provided. The topic is developed in a way that can be suitable for both undergraduate student and general physicist.

An account of peculiar light patterns produced by reflection in a pool under falling rain droplets was recently reported by Molesini and Vannoni (2008 Eur. J. Phys.29 403–11). The mathematical approach, however, only covered the case of a symmetrical location of a light source and the observer's eyes with respect to the vertical of the falling droplet. Extending such an account to asymmetrical configurations, a general solution for the reflection trails that are observed on the pool surface is presented.

In this paper, we consider solutions to the three-dimensional Schrödinger equation of the form ψ(r) = u(r)/r, where u(0) ≠ 0. The expectation value of the kinetic energy operator for such wavefunctions diverges. We show that it is possible to introduce a potential energy with an expectation value that also diverges, exactly cancelling the kinetic energy divergence. This renormalization procedure produces a self-adjoint Hamiltonian. We solve some problems with this new Hamiltonian to illustrate its usefulness.

The Boltzmann factor is the basis of a huge amount of thermodynamic and statistical physics, both classical and quantum. It governs the behaviour of all systems in nature that are exchanging energy with their environment. To understand why the expression has this specific form involves a deep mathematical analysis, whose flow of logic is hard to see and is not at the level of high school or college students' preparation. We here present some experiments and simulations aimed at directly deriving its mathematical expression and illustrating the fundamental concepts on which it is grounded. Experiments use easily available apparatuses, and simulations are developed in the Net-Logo environment that, besides having a user-friendly interface, allows an easy interaction with the algorithm. The approach supplies pedagogical support for the introduction of the Boltzmann factor at the undergraduate level to students without a background in statistical mechanics.

The flight trajectory of a water rocket can be reasonably calculated if the magnitude of the drag coefficient is known. The experimental determination of this coefficient with enough precision is usually quite difficult, but in this paper we propose a simple free-fall experiment for undergraduate students to reasonably estimate the drag coefficient of water rockets made from plastic soft drink bottles. The experiment is performed using relatively small fall distances (only about 14 m) in addition with a simple digital-sound-recording device. The fall time is inferred from the recorded signal with quite good precision, and it is subsequently introduced as an input of a Matlab® program that estimates the magnitude of the drag coefficient. This procedure was tested first with a toy ball, obtaining a result with a deviation from the typical sphere value of only about 3%. For the particular water rocket used in the present investigation, a drag coefficient of 0.345 was estimated.

It is shown how a modern extension of Fourier analysis known as wavelet analysis is applied to signals containing multiscale information. First, a continuous wavelet transform is used to analyse the spectrum of a nonstationary signal (one whose form changes in time). The spectral analysis of such a signal gives the strength of the signal in each frequency as a function of time. Next, the theory is specialized to discrete values of time and frequency, and the resulting discrete wavelet transform is shown to be useful for data compression. This paper is addressed to a broad community, from undergraduate to graduate students to general physicists and to specialists in other fields than wavelets.

The cooling of objects is often described by a law, attributed to Newton, which states that the temperature difference of a cooling body with respect to the surroundings decreases exponentially with time. Such behaviour has been observed for many laboratory experiments, which led to a wide acceptance of this approach. However, the heat transfer from any object to its surrounding is not only due to conduction and convection but also due to radiation. The latter does not vary linearly with temperature difference, which leads to deviations from Newton's law. This paper presents a theoretical analysis of the cooling of objects with a small Biot number. It is shown that Newton's law of cooling, i.e. simple exponential behaviour, is mostly valid if temperature differences are below a certain threshold which depends on the experimental conditions. For any larger temperature differences appreciable deviations occur which need the complete nonlinear treatment. This is demonstrated by results of some laboratory experiments which use IR imaging to measure surface temperatures of solid cooling objects with temperature differences of up to 300 K.

We examine the equant model for the motion of planets, which was the starting point of Kepler's investigations before he modified it because of Mars observations. We show that, up to first order in eccentricity, this model implies for each orbit a velocity, which satisfies Kepler's second law and Hamilton's hodograph, and a centripetal acceleration with an r⁻² dependence on the distance to the Sun. If this dependence is assumed to be universal, Kepler's third law follows immediately. This elementary exercise in kinematics for undergraduates emphasizes the proximity of the equant model coming from ancient Greece with our present knowledge. It adds to its historical interest a didactical relevance concerning, in particular, the discussion of the Aristotelian or Newtonian conception of motion.

We show that the underlying mathematical structure of dimensional analysis (DA), in the qualitative methods in problem-solving context, is the algebra of the affine spaces. In particular, we show that the qualitative problem-solving procedure based on the parallel decomposition of a problem into simple special cases yields the new original mathematical concepts of special points and special representations of affine spaces. A qualitative problem-solving algorithm piloted by the mathematics of DA is illustrated by a set of examples.

In this paper we study an acoustic system comprised of two identical closed tubes connected with a short one, and a quantum-mechanical square well with a delta function potential. Both systems can be thought of as mathematically equivalents to the homogeneous string with a concentrated mass at its middle point. We describe also a simple experimental device to measure the resonances of the acoustic system as a function of the connecting tube section. The experimental setup and the theoretical approach we have employed can be used in both undergraduate and graduate level courses. Moreover, this work may be useful as a teaching aid in other experimental research projects such as for instance the study of the emergence of the band structure in solid state physics.

The teaching of physics through practical experiments has long been an established practice. It forms a key component of teaching of that subject at both school and university levels. As such, students have strong views of this method of teaching. This paper reports on the view of undergraduate physics students in relation to their experiences of practical physics at school. 500 students across three Higher Education Institutions in the UK were surveyed to determine their perceptions, views and opinions in this area. This paper initially presents the overall views of the students, and then looks in more detail at the effect the different levels to which students took the subject at school affected those views. Specifically, students who took Advanced Higher versus Higher are compared, as well as those who took Advanced Higher versus A-level. Comparison was also made between the responses of female and male students. The general picture is very encouraging, with students broadly appreciating the practical side of physics.

Expressions for the chemical potential of an Einstein solid, and of ideal Fermi and Bose gases in an external one-dimensional oscillatory trap, are calculated by two different methods and are all found to share the same functional form. These derivations are easier than traditional textbook calculations for an ideal gas in an infinite three-dimensional square well. Furthermore, the results indicate some important features of chemical potential that could promote student learning in an introductory course in statistical mechanics at the undergraduate level.

The relation between the symmetry properties of a physical system and conservation laws, although a fundamental topic, is barely treated in educational contexts. In this paper, by only using elementary formalism, we explore the connection between energy–momentum conservation laws with the reference frame invariance either in the Galilean or in the Lorentzian context, which is a fundamental symmetry property of any physical system. Some reflections (pedagogically sound and significant either for graduate and university undergraduate students) are also proposed on the different roles played by space and time in pre-relativistic and relativistic mechanics.

This paper describes a simple noise circuit for the undergraduate physics laboratory. Students use this circuit to study the properties of electrical noise on a personal computer. This is made possible by using a data acquisition system that allows the experimenters to obtain large amounts of data on the computer, suitable for subsequent mathematical computations. Various properties such as mean, noise power, noise power density and the probability distribution of noise voltages are also explored.

At the beginning of academic year 2007–08, staff in the Department of Physics and Astronomy at the University of Glasgow started to implement a number of substantial changes to the administration of the level 1 physics undergraduate class. The main aims were to improve the academic performance and progression statistics. With this in mind, a comprehensive system of learning support was introduced, the main remit being the provision of an improved personal contact and academic monitoring and support strategy for all students at level 1. The effects of low engagement with compulsory continuous assessment components had already been observed to have a significant effect on students sitting in the middle of the grade curve. Analysis of data from the 2007–08 class showed that even some nominally high-achieving students achieved lowered grades due to the effects of low engagement. Nonetheless, academic and other support measures put in place during 2007–08 played a part in raising the passrate for the level 1 physics class by approximately 8% as well as raising the progression rate by approximately 10%.

Computer-assisted experiments with a two-wire transmission line are described. The use of a data-acquisition system makes the measurements more convenient and less time consuming; therefore, additional exercises are added to the traditional student experiments. The characteristic impedance of the line is determined. The zero input impedance of the quarter-wave open-circuited line and the half-wave short-circuited line is demonstrated. Any unknown impedance loading the line can be determined by observing the standing wave.

A simplified version of the Curzon–Ahlborn engine is proposed by assigning the same thermal resistance and same temperature difference at the upper and the lower isotherms of the Carnot cycle. The value of efficiency at the maximum power matches for all the three real heat engines reported by them but at the cost of marginally lower power output. The results are obtained in closed form and are easily reproducible by undergraduates using the componendo and dividendo rule and simple differentiation in contrast to considerable algebra and partial differentiation in the Curzon–Ahlborn derivation.

We present the analytic solutions of the saturated absorption spectrum for an atomic medium composed of two-level atoms and describe the effects of saturation and optical pumping on the spectrum. The background-subtracted Doppler-free spectrum is Lorentzian-shaped with an amplitude and width given by 1 − {1 + s₀[1 + (γ₂t/2)]}^−1/2 and γ_t[1 + {1 + s₀[1 + (γ₂t/2)]}^1/2], respectively, where γ_t is the transverse decay rate, γ₂ is the decay rate to energy levels other than the ground state, s₀ is the saturation parameter and t is the interaction time of the atoms with laser beams. Analytic results are compared with numerical results and show good agreement with one another. This paper is intended for graduate students who have started to study basic atomic physics.

We present an experiment for education which demonstrates random transmission or reflection of heralded single photons on beam splitters. With our set-up, we can realize different quantum random experiments by appropriate settings of polarization rotators. The concept of entanglement is motivated by correlated randomness. The experiments are suitable for undergraduate education and are available as interactive screen experiments.

An expression is developed for the energy dissipated when a constant external force is suddenly applied to the end of a particle moving in 1D subject to a conservative restoring force and a general damping force. Assuming the particle was initially at rest and ends up in static equilibrium, the fraction of the mechanical energy lost depends only on the conservative force and on the net displacement of the particle. Mechanical models are suggested to illustrate the ideas and their capacitor circuit analogies. The treatment is appropriate for an intermediate-level undergraduate mechanics course.

We show that the virial theorem provides a useful simple tool for approximating nonlinear problems. In particular, we consider conservative nonlinear oscillators and obtain the same main result derived earlier from the expansion in Chebyshev polynomials.

The usual 'quantum bouncer' is a quantum particle bouncing elastically under the influence of gravity. Here we consider a particle bouncing off an impenetrable wall, and bound to it by the harmonic potential instead of by the gravitational potential mgx. An analytic solution is possible, easily accessible to classes in intermediate quantum mechanics. The solution is related to Schrödinger's oscillating wavepacket (the original coherent state). Although periodic, it changes its form during the oscillation, in contrast to Schrödinger's packet. The uncertainties in the position (Δx) and the momentum (Δp) oscillate also, as does the product ΔxΔp.

We show that an improved solution to the carousel-pendulum problem can be easily obtained through a first-order Taylor expansion, and its accuracy is determined after the obtention of an unusable analytical exact solution, advantageously replaced by a numerical one. It is shown that the accuracy is unexpectedly high, even when the ratio length of the pendulum to carousel radius approaches unity.

In a recent letter, Beléndez et al (2009 Eur. J. Phys.30 L25–8) proposed an alternative of approximation for the period of a simple pendulum suggested earlier by Hite (2005 Phys. Teach.43 290–2) who set out to improve on the Kidd and Fogg formula (2002 Phys. Teach.40 81–3). As a response to the approximation scheme, we obtain another analytical approximation for the large-angle pendulum period, which owns the simplicity and accuracy in evaluating the exact period, and moreover, for amplitudes less than 144° the analytical approximate expression is more accurate than others in the literature.

In their comment, Yuan and Ding derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation for the simple pendulum motion. However, we point out that another procedure is possible to obtain an approximate expression for the period. This procedure is based on the approximation of the exact period formula—which is expressed in terms of a complete elliptic integral of the first kind—instead of the approximation of the original differential equation. This last procedure is used, for example, by Carvalhaes and Suppes using the arithmetic–geometric mean.

The didactic approach to Noether's theorem in classical mechanics presented in Marinho Jr (2007 Eur. J. Phys.28 37–43) is re-examined.

This reply answers the issues raised in the comment on my paper (Marinho Jr 2007 Eur. J. Phys.28 37–43), obtains the Laplace–Runge–Lenz vector (Goldstein 2002 Classical Mechanics 3rd edn (Reading, MA: Addison-Wesley)) using Noether's theorem and includes a Maple program used to derive the results.