Table of contents

This paper suggests a new way of computing the path integral for simple quantum mechanical systems. The new algorithm originated from previous research in string theory. However, its essential simplicity is best illustrated in the case of a free non relativistic particle, discussed here, and can be appreciated by most students taking an introductory course in quantum mechanics. Indeed, the emphasis is on the role played by the entire family of classical trajectoriesin terms of which the path integral is computed exactly using a functional representation of the Dirac delta distribution. We argue that the new algorithm leads to a deeper insight into the connection between classical and quantum systems, especially those encountered in high-energy physics.

The generally hidden phenomenon of optical polarization plays an ever increasing part in our scientific as well as daily lives. Although it is usually treated primarily in advanced physics courses, many phenomena could well be taught at a much more elementary level. After enumerating some of the many uses in science and technology, and looking at the most easily prepared polarization phenomena, we discuss Haidinger's brush, a butterfly-like image which can be observed in the sky with the naked eye alone.

The Fermi contact term of the hyperfine interaction is derived with a minimum of quantum mechanical assumptions and the use of two different length scales. The latter approach clarifies the mechanism, which has traditionally been subsumed by the term `contact'.

A simple and illustrative rheonomic system is explored in the Lagrangian formalism. The difference between the Jacobi integral and the energy is highlighted. A sharp contrast with remarks found in the literature is pointed out. The non-conservative system possesses a Lagrangian that is not explicitly dependent on time and consequently there is a Jacobi integral. The Lagrange undetermined multiplier method is used as a complement to obtain a few interesting conclusions.

We propose an elementary approach for introducing the quasi-classical quantum states of a spinless charged particle in a uniform magnetic field. By exploiting the similarity with the case of the two-dimensional harmonic oscillator as well as a property of the solutions of the pertinent Schrödinger equation, we derive two basic solutions. One of them is a stationary, minimum-uncertainty wavepacket centred at an arbitrary point. This corresponds to a classical particle at rest. The other solution is a minimum-uncertainty wavepacket that rotates at the classical angular speed. The expected value of the energy agrees with the classical prediction within the zero-point energy. These results are obtained without any knowledge of the energy eigenstates. An appendix suggests a brief and self-contained procedure for writing the Hamiltonian of a charged particle under the Lorentz force.

A simple diagram is presented for use with the analysis of the Dirac equation for an atom with a single electron. The complicated expression for the energy of the electron takes on a very simple form. The usual theory is presented with an angle from the diagram as the energy parameter, and the radial parts of the energy eigenfunctions for n= 1 and 2 are calculated.

Students often have difficulty understanding the concepts of entropy and irreversibility, and to a lesser extent, temperature. This is partially due to the statistical nature of these concepts and the abstract connection between probability and energy. The example of a large collection of coins is used to elucidate the basic concepts of probability (in particular, the law of large numbers), and uses them in the same setting to disentangle the more difficult notions of temperature, entropy, and irreversibility.

The Karlsruhe Physics Course is an attempt to modernize the physics syllabus by eliminating obsolete concepts, restructuring the contents and extensively applying a new model, the substance model. The course has been used, tested and improved for several years, and we believe that the time has come to make it known to a wider public. We introduce the structure which underlies the course and discuss some consequences for the teaching of various subfields of physics.

The error resulting from using the well known paraxial approximation instead of Snell's law of refraction is visualized geometrically.

We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.

The basic properties of momentum distributions in quantum mechanics for elementary systems as well as their (semi-)classical analogue derived in a preceding paper are illustrated by a detailed analysis of the Morse oscillator and the three-dimensional Coulomb potential, where the distributions can be calculated in closed form.

A simple way of deriving the electromagnetic boundary conditions in the differential-form formalism is outlined. The derivation is based on glueing together two independent electromagnetic source-field systems, each existing in an arbitrary electromagnetic environment. The combined electromagnetic source-field system contains a surface separating regions of the two original systems and media. It is shown that, in general, additional sources at the surface are needed for the new system to satisfy the Maxwell equations. This requirement is seen to create a set of boundary conditions at the surface. As a by-product, Huygens' principle can be simply formulated in differential form formalism.

We address the problem of finding the Euler angles (a,b,c) connecting two Cartesian frames which are rotated infinitesimally (in the most general sense) with respect to each other. It is pointed out that while a + cand bare first-order small quantities, the angles aand cshould be of order unity.

The heating of simple geometric objects immersed in an isothermal bath is analysed qualitatively through Fourier's law. The approximate temperature evolution is compared with the exact solution obtained by solving the transport differential equation, the discrepancies being smaller than 20%. Our method succeeds in giving the solution as a function of the Fourier modulus so that the scale laws hold. It is shown that the time needed to homogenize temperature variations that extend over mean distances x_mis approximately x_m²/, where is the thermal diffusivity. This general relationship also applies to atomic diffusion. Within the approach presented there is no need to write down any differential equation. As an example, the analysis is applied to the process of boiling an egg.

The thermoglobe is a new instrument intended for use in schools for teaching the origin of the seasons. This globe changes its colour to red at those parts which are heated. When the rotating thermoglobe is irradiated by a heating lamp, a pattern of red colour is produced that corresponds to the seasonal temperature distribution. This teaching strategy shows clearly that the seasons are due to the inclination of the terrestrial axis with respect to the orbital plane.

It is demonstrated that traditional quantization of the Dirac equation also introduces the Dirac sea into quantum electrodynamics. A discussion of the Dirac equation with a potential shows that the symmetry of charge conjugation leads to a formal ambiguity of its solutions, which can be made use of to give a physical meaning to apparently unphysical solutions. In this way it seems possible to do without the introduction of the Dirac sea. The procedure can be generalized to many particles, i.e. field theory in the shape of relations between certain matrix elements of the field operators. The application of these relations is demonstrated in a simple example. Finally it is shown how in this alternative formalism pair creation can be understood without the Dirac sea.

It is pointed out that the distinction between `standard' and `non-standard' representations of the radial delta function (r) emphasized by Menon in a recent paper on the solving of the radial Poisson equation for a point charge is devoid of any significance. It is also shown how the solution 1 / | r- r´ | of the three-dimensional Poisson equation for a point charge can be derived with no recourse to any representation of the delta function, thereby clarifying the solution's precise meaning.

Evidence is presented which corrects the conclusions of a paper in this journal (Topper D and Vincent D E 1999 Eur. J. Phys2059 - 66) that Newton lacked a full understanding of the motion of a projectile near the surface of the Earth.

We reply to a criticism of our paper (1999 Eur. J. Phys.2059-66) made by Michael Nauenberg.

A basic flaw is identified in Dodd's (1983) explanation of the Compton effect within classical electrodynamics.

The authors of this textbook are two experimental physicists working in condensed matter who state that they wrote it looking for a pragmatic approach. With this premise the book could be expected to concentrate on the applications at the expense of the principles. This is not the case, however, the authors having achieved a good balance.

The text is intended for the second part of a degree course, assuming that the students have a good grasp of calculus and classical physics, especially of classical mechanics and electromagnetism, and that they have some familiarity with the main ideas of modern physics. For that level, the book can be described as advanced but not as excessive.

The first five chapters, covering 200 pages out of the total of 474, are devoted to the postulates, which could seem too much for a pragmatic approach. But the authors have chosen to present the mathematical background and the physical consequences of quantum mechanics as soon as possible in association with the postulates. It is their way to reach a balance between principles and applications. Although they rely on intuition, they care for the precision and develop the calculation with detail. This, their effort to relate the applications to the postulates, is perhaps the most valuable aspect of the book.

In my opinion, the method works. For instance, the Heisenberg relations are introduced following the first postulate, before presenting the idea of an operator associated with a dynamical variable, with a description of the Fourier transform, although explaining with care the idea of uncertainty. The concepts of Hermitian and unitary operators are introduced in two examples in the chapter on the second postulate, and so on. Although some physicists would prefer a more clear-cut separation between mathematical prerequisites, postulates and consequences, especially the theorists (of whom I am one), the book will certainly be effective and suitable for general students.

The second part develops the applications of the postulates to physical problems, with chapters on one-electron atoms, angular momentum, approximate methods, many-particle systems, and atomic and nuclear radiation.

The book contain a good number of exercises and problems, carefully solved in detail, which should allow the students to do their homework with useful assistance.

To summarise: this is a good and recommended textbook for a course on quantum physics, intended for a pragmatic preparation to the study of atoms, nuclei or condensed matter. It could be suitable also as a first introduction, prior to deeper analysis, for those who aim to understand later the frontiers of quantum theory itself.

There is an old saying which goes `Don't judge a book by its cover'. This could be modified for Professor Dahl's book to state `Don't judge a book by its title'. For deals not only with the physics but puts the whole process in the context of history. It starts with the events of 1919, both in Manchester and Paris, and leads on to the eventual discovery of the neutron. The first clear indication of the existence of an isotope of hydrogen appeared in 1922, although as early as 1913 Lamb and Leen of New York University were aware that `there was something amiss' with the composition of water. Work continued until, in the Spring of 1933, pure heavy water was first produced. The chapters of the book follow on in chronological order starting with artificial radioactivity, leading on to the ultimate goal of nuclear fission. Within this, of course, the pivotal role played by heavy water in the process takes centre stage, with much attention being given to the production process by Norsk Hydro.

The key role to be played by heavy water was well recognised by scientists in England, France and Germany. There is an excellent graphic account of the efforts to bring the stock of heavy water from Norway to France via Scotland before Norway was invaded. Efforts by the Allied forces to destroy the heavy water production plant at Vemork, under codenames such as Freshman and Gunnerside, make interesting reading, as does the account of the attack undertaken to prevent the remaining stocks being shipped back to Germany.

Overall it is a book that is well presented, with good diagrams and historic photographs of the main players and locations. It is well annotated and has a useful series of appendices. If I have a minor quibble then it is with the author when he introduces a new personality to the narrative and then tends to backtrack to give an account of that person's life. This tended to spoil the flow of the narrative for me.

The essence of a good book is that not only is it interesting but when you put it down you will have learned something. I found that Professor Dalh's book met both criteria and was an enjoyable read.

In the past few years there has been an explosion of books on chaos: the `strange' behaviour of nonlinear dynamical systems. Our understanding of nonlinear dynamics has been aided by the availability of computers with graphical displays on which one can study the iteration of maps and the solution of ordinary differential equations (ODEs). This book is a useful addition to the literature.

The author is an academic mathematician, and has developed this book from lecture notes for courses he has given, which involved some laboratory work, based on computer programs. The book expects the reader to have access to the author's programs, which can be obtained from his web site (sunsite.anu.edu.au/education/chaos ) in a package Chaos for Java . This can be accessed directly using a Java enabled browser. His use of Java means that the programs should be more or less independent of the computer on which they are running provided one has access to an up to date version of a Java Virtual Machine (which may be downloaded, free, from the web for many computers).

The author has told me that, while he does not propose to make his source code available, he plans, when the application version is up and stable, to factor the code which computes maps and ODEs so that users can write their own extensions, for other maps and ODEs, in a way which will be picked up automatically provided the Java class files are in the same directory (one can get the Java class loader to look for other classes).

I have tried out some of his software which I found easy to use. There are instructions in the book, but the programs have been written so that one can run most of them intuitively, without referring to the book for detailed instructions. The programs have nice features; for example, when one is varying an angle continuously it increases smoothly through 360°.

The author's FFT program allows the number of sample points to be any number which is a multiple of powers of the primes 2, 3, 5, 7, 11 and 13. Thus one is not limited to 2ⁿsample points. The value of this extra freedom is well demonstrated in examples where he demonstrates a period-3 orbit of the logistic map, and a period-7 window of the Hénon map.

Lyapunov exponents and Poincaré sections are discussed and are explained with clarity.

The book is mathematical in its approach. In the first three chapters it considers one-dimensional maps, mostly the tent and logistic maps, in considerable detail. Chapter 4 covers two-dimensional maps, chapter 5 fractals and chapter 6 some simple ordinary differential equations.

This is a book which ought to be in every physics library (and, of course, in every maths library). Anyone wanting to obtain a deeper understanding of chaos, or presenting a course on it, ought to think about buying the book. It is suitable for an undergraduate course in computational mathematics or theoretical physics. It does not contain many physical examples, so might be less attractive than some other texts for a course on chaos for students of experimental physics.

Dear Reader,

The Editorial Board of European Journal of Physicsmeets once a year in order to discuss journal policy, and here I will outline some of the matters which arose at our meeting in August 1999.

Our third Special Issue was published in November 1999 with the title `Unsolved Problems in Physics'. Will the new millennium, just started, soon provide the solution to some of these problems? We do hope so! However, we do not assume that the search for new knowledge in physics will ever come to an end. Our Special Issue for 2000 will have the title `Quantum Physics in Solids'. The `quantum' in the title marks the centenary of quantum physics. Some invitation letters for papers have already been sent out but you are all very welcome to contribute.

The confusion amongst authors between our journal and its similarly named counterpart covering research in physics remains. Also the educational level of the papers for our journal has been questioned. Therefore the Board has agreed to include `university-level education' in the subtitle of European Journal of Physicsto guide our authors. However, we urge all our authors to read carefully the journal's scope and guidance for authors, where it is requested that article introductions should indicate the relevance of the content to university physics teaching.

Experimental articles are always welcome and some student project work may be suitable for European Journal of Physics . These articles might be accepted even if they cannot be shortened sufficiently to meet our usual requirement for the length of articles. We may introduce a new heading `Laboratory Workshop' if we receive enough of these papers. In some cases, supplementary material can be placed in an online appendix accessible via the electronic journal. Articles on the use of videos in teaching might stimulate a debate on the merits of this type of enhancement.

Comments on published articles are very useful for our readers. Such Comments should be short and normally occupy no more than two pages. When the latest update to the electronic journals system becomes operational during this year it should be possible to include a forward link between a particular paper and any Comment published later.

A new electronic author and referee service is being developed for the Institute of Physics Publishing website, enabling both authors and referees to discover the fate of each paper, rather than having to make enquiries via the Editorial Office.

Finally, my term of office as Honorary Editor of European Journal of Physicsis now coming to an end and I want to express my thanks to all our authors for their very stimulating contributions to the journal during that time.

With best wishes