Table of contents

Using a semi-quantitative variational strategy, we deduce the form of the characteristic length scale of several bound quantum many-body systems with forces of the 1/r² type. The difference resulting from considering fermions or bosons is emphasized.

Many problems of physical interest - for instance, in statistical mechanics - are described by linear ordinary second-order differential systems for which different types of transfer matrices can be introduced and used. Focusing on heterostructures where matching at interfaces is involved, this paper discusses two of them with emphasis on one, here denoted T, which involves the linear differential form expressing the physical quantities matched at the interfaces. The mathematical background is summarized in a simple way and then T is used to study two types of heterostructures involving a large number of interfaces. Firstly, the regular periodic superlattices are studied and the role of different boundary conditions (BCs) at the end of one period is discussed. Only periodic BCs are suitable to study a simple regular superlattices but the discussion provides the background to study different approximants when the period is a largish generation of a quasi-regular heterostructure, like, for instance, a Fibonacci sequence.

In this paper we review and discuss the approaches used, almost universally, in textbooks dealing with quantum mechanics, and particularly those focused on optoelectronics devices, to explain the atom-field interactions. For this purpose, a true understanding and careful use of the first-order perturbation theory are necessary. By providing two alternative full derivations of the absorption/emission processes when the radiation is in a coherent multimode state, we highlight a number of conceptual and didactical failures in the usual textbook presentations, and propose more suitable and convincing strategies to improve them.

The well-known soliton potentials, as defined on a bounded interval of the real line and continued beyond by periodicity, are considered. In such a way one obtains new exactly solvable periodic potentials for the Schrödinger equation. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Kronig-Penney potential. Solutions of the Schrödinger equation are expressed in terms of elementary functions. The potentials involved might become non-standard textbook examples with possible applications in solid-state physics.

The equations of the most general type of solar calendar and sundial, which were derived in Vandyck M (2000 A unified and general framework for sundials and solar calendars Eur. J. Phys.22 79), are extended here to take into account the three main systems of time-reckoning employed by the Ancients: Babylonian hours, Roman hours, and Italian (or Bohemian) hours. The results provide insight into the notions of `temporary' and `equal' hours, which are related to the concept of `uniformity' of the flow of time.

This paper describes the absolute temperature scale of Kelvin (William Thomson). Kelvin found that Carnot's axiom about heat being a conserved quantity had to be abandoned. Nevertheless, he found that Carnot's fundamental work on heat engines was correct. Using the concept of a Carnot engine Kelvin found that Q₁/Q₂ = T₁/T₂. Thermometers are not used to obtain absolute temperatures since they are calculated temperatures.

We consider exactly solvable models in quantum mechanics with potential-energy functions that exhibit exponential dependence on the coordinate. As particular examples we choose a spherically symmetric well to illustrate the concept of critical strengths, and a barrier in one dimension to discuss tunnelling.

The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator is true only in one or three dimensions. In general, an extra term of the form ℏ²(n-1)(n-3)/2m·4r² has to be added to the Hamiltonian.

We characterize and classify the quantum states of a pair of spin-½ particles in terms of the entanglement of the pair. We describe a general strategy for classifying the quantum states of entangled systems, and apply it to the simplest entangled system available - a pair of spin-½ particles. The method is based on local unitary transformations. We give a set of six parameters that describe an arbitrary pure state of a pair of spins, one of which is the entanglement of the pair. We discuss visual representations of the state of this system, and propose a visualization using a Bloch sphere for each subsystem and a disc to record the `shared' information.

At present, the global positioning system (GPS) is the best way for worldwide time transfer, in all the scientific and practical applications, both for its accuracy and reliability. Evidence of this is also available to university students in physics, through simple experimental activities, today made possible by the low cost of GPS modules dedicated to timing. The 1PPS (one peak per second) pulse, made available by such receivers, brings into a teaching laboratory the stability of an ensemble of atomic clocks, capable of very accurate frequency (which we have described previously) and time measurements.

In fact, for the last year, we have also utilized the 1PPS pulse as the time reference for a time server, suitable for distributing accurate and reliable time information through the internet. This paper shows the learning opportunities we have found for understanding how a remote clock synchronization can be achieved, even in an open, non-dedicated network like the internet. The combination of a modern measuring tool, like the GPS, and the easy access through the internet to time servers spread around the world, make it possible to extract useful information from the comparison of one server against the others.

Direct activities by students, on setting-up their computer for a special application, making actual measurements with the instruments available in the teaching laboratory, verifying in real time the typical performance of the network and so on, have been a trigger for their interest. We feel that such exercises can help them to gain a deeper knowledge of these modern instruments, in spite of their advertised utilization only as `black boxes', too sophisticated to be clearly understood.

We present a simple experiment, specifically designed for students of undergraduate optics courses, where the influence of an aperture stop position on the three-dimensional structure of the focal volume of focusing systems is studied. The experiment, which involves only simple optical elements, permits an undergraduate student to generate different focal structures by simply axially displacing the aperture stop.

Cosmology is an attractive subject for students but it is usually difficult if general relativity is not understood. In this article, we first recall the Newtonian derivation of the Friedmann equations which govern the dynamics of our universe and discuss the validity of such a derivation. We then study the equations of evolution of the universe in terms of a dynamical system. This sums up the different behaviours of our universe and enables us to address some cosmological problems.

A method for integrating energy conversion phenomena, nonlinear circuit behaviour and hysteresis into a simple laboratory activity is demonstrated. Current-voltage data for a flashlight bulb driven at various frequencies and amplitudes are measured and simulated. Non-ohmic behaviour and hysteresis are observed experimentally and reproduced numerically with a model that includes the temperature-dependent specific heat capacity and electrical resistivity of the filament. The phenomenological model and simple experimental design could be modified for a variety of audiences.

Fractional revivals of wave packets in an infinite square well are scrutinized with a viewpoint rooted in Fourier analysis, and a compact relation, expressing the wavefunction Ψ(x,t) for certain values of t in terms of spatially displaced copies of Ψ(x,0), is derived without appealing to a classical analogy; conditions for the appearance of an interference pattern in a plot of the position probability density of a wave packet are deduced, along with related results obtained previously from different considerations, with minimal effort. The specific case of a packet of Gaussian shape is analysed in greater detail to provide concrete illustrations of fractional revivals with or without interference between overlapping copies of wave packets, depending on whether a finite value of the so-called classical time can or cannot be assigned to the wave packet.

Monte Carlo simulations are an important tool in modern-day studies of many physical systems. Where unlikely events are to be simulated, the importance sampling technique can considerably ease the processing burdon, without compromising statistical significance. Here a comparison of importance sampling and standard Monte Carlo simulations is given. Emphasis is on variance reduction, and on the simulation gain of importance sampling, which is calculated explicitly for a simple example.

Much controversy exists about the supposed changing examination standards. Emphasis has been placed on the standards of GCSE and A-level examinations. However, many large employers recruit graduates, and so university examination standards also deserve attention. Here, Cambridge University Part II (third year undergraduate) examinations in Physics are studied since 1940. Trends in prescriptiveness, choice of questions, and other variables were found.

Medical physics is currently a rapidly growing field of physics. Numerous academic, clinical and industrial opportunities are open to physicists in the medical world. I report on an intermediate level physics course on medical physics taught at Carleton College. The topics covered in this course cover all areas of physics, but with examples drawn from medical applications. In addition to physics majors, this course appeals to biology, chemistry and pre-medical students who have a keen interest in physics.

This paper presents the Ashby and Jephcott laser interferometer that is quite simple to set up. This interferometer is well known in the field of plasma physics and has been employed to measure the electron density in transient ionized gases (plasmas). It is used here to determine both the amplitude and the averaged speed of a loudspeaker membrane moving at a low frequency (500 Hz).

The paper describes the behaviour of a simple mechanical system, which should help students (or teachers) to understand and clarify the importance of relative motion of two surfaces when kinetic friction is present. We show that despite the simplicity of this system, the peculiar interplay between friction forces, tension forces and gravity leads to physical solutions exceeding in many cases most intuitive expectations. These are discussed in detail. The problem is intended to be solved in a theoretical framework as an example, which helps to understand better the physical background of kinetic friction phenomena.

We consider the motion of a particle subjected to a general rheonomic contraint, discussing the conservation conditions of the mechanical energy and of the Langrangian energy from the Lagrangian point of view. We generalize the results obtained in an article recently published in this journal (de Castro A S 2000 Eur. J. Phys.21 23-26).

The charge of an electron and the permittivity of the vacuum, when expressed in the natural system of units, are proportional with the dimensionless factor 2α where α is the fine structure constant. Therefore, the electronic charge and the permittivity of the vacuum can be fundamental constants on equal footing, but not simultaneously.

In the literature of physics Hendrik Casimir features amongst a multitude of names associated with Laws and Effects, each name a convenient label for a precise physical relationship of one kind or another. Such names generally connote the provenance of the scientific principles they convey, usually identifying the individual or individuals accredited with the first public pronouncement or publication. The history of science is of course replete with examples of how societal criteria have borne upon the attachment and acceptance of such names, especially where the science has been the subject of more than one individual's work. However this is not so in the case of Casimir, whose ground-breaking work on microscopic forces has been granted eponymy unequivocally.

To mark the sad news of his death last year, this special section of European Journal of Physics celebrates the discovery, underlying principles and implications of Casimir forces, together with a biographical sketch that helps illustrate the man's much wider sphere of scientific influence. In publications that spanned 45 years his research output covered an astonishing range of topics in physics, and in a sense the necessary focus here on one particular branch of his work does a necessary disservice to the whole. Nonetheless there are special reasons for focussing on his work on microscopic forces, because of their highly significant implications for the foundations of physics. The experimental verification of the results Casimir obtained has been widely interpreted as vindicating some of the most important but least intuitive principles of quantum electrodynamics, the progeny of quantum mechanics and special relativity.

Casimir was a man highly interested in the communication of science at all levels, from conferences to magazines for the general public, and with a genuine interest in science-related philosophical issues. A popular speaker at both formal and informal occasions, he was also a keen and able writer, serving for several years on the Editorial Board of the Dutch literary magazine De Gids. Within and beyond the Philips organisation Casimir actively promoted European collaboration in science; he also increasingly directed efforts towards the advancement of professional standards in research management. With all these interests and activities, and also as a former President of the European Physical Society (under whose auspices this journal is published) it is particularly appropriate to mark his passing with the publication of this commemorative section of European Journal of Physics. It has been my great pleasure and honour to be involved.

I would like to thank all who have contributed to the special papers for this issue. I am especially pleased to acknowledge very helpful communications with the Philips organisation, and in particular Lou-Fe' Feiner, whose enthusiasm for this project afforded me valuable personal recollections and many other useful insights. For further information the interested reader will undoubtedly appreciate reading Casimir's autobiography Haphazard Reality. Half a Century of Science 1983 (New York: Harper and Row).

Note added in proof: As this edition was going to press, news also emerged of the recent death of Dik Polder, whose collaboration with Casimir led to some of the key results celebrated in this issue. It is fitting to record his important contributions to the work, and to extend tribute to a man whose name will remain inextricably linked with Casimir's.

If one checks various dictionaries for the name `Casimir', one will find quite a few entries under the subjects of mathematics and physics, as well as in many international Who's Whos. All refer to the same person, who is particularly associated with a mathematical concept, the `Casimir operator', and a physical phenomenon, the `Casimir effect'.

The origins and physical significance of the Casimir effect are reviewed, linking the zero-point energy of the vacuum in quantum electrodynamics with a force between conducting plates. It is shown how, by the use of dimensional and other simple physical arguments, the major features of the phenomenon can be derived.

The Casimir-Polder potential is introduced in a brief historical discussion of intermolecular forces. To show how the potential can be found from basic ideas of elementary physics, the interaction energy between two polarizable molecules is first found in classical electrodynamics when the molecules are bathed in a non-uniform and time-varying external field. This result is then used to find the Casimir-Polder interaction between the pair by taking the external field to be the vacuum field in quantum electrodynamics. An alternative route to obtain the Casimir-Polder potential is outlined. This involves examining the change in the zero-point energy of the radiation field when two polarizable molecules are placed within a large empty cavity. Furthermore, it is shown how many-body forces are endemic in any situation where the forces are mediated by the electromagnetic Maxwell fields. In particular, the three-body energy shifts that occur when three molecules are present is found using the method of zero-point energy differences.

We will describe some aspects of Casimir effects that appear to be of particular significance now, more than half a century after Casimir's famous paper.