Volume 58, Issue 12, December 1990
 Papers


Bell’s theorem without inequalities
View Description Hide DescriptionIt is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two‐particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin‐correlation and multiparticle interferometry examples are given of suitable three‐ and four‐particle arrangements, both at the gedanken and at the real experiment level.

Electron wavelike behavior: A historical and experimental introduction
View Description Hide DescriptionFollowing the Fresnel theory of light and the consequent scientific debate that his formulation generated, two key experiments with electrons, diffraction by means of a circular hole and a circular obstruction, have been realized to show the existence of the Fresnel zones and of the so‐called ‘‘Poisson spot.’’ The basic arguments concerning the quantum mechanical nature of electrons can be introduced by taking advantage of the vivid impression stimulated by the experimental images.

Lindhard’s paradox—Diffusion of magnetic field into a perfect conductor
View Description Hide DescriptionThis paper develops the argument, presented briefly by Lindhard in 1953, that a perfectly conducting metallic sample is unable to keep out a steady magnetic field, since even in the absence of collisions the field energy can still be reduced by transfer to the electron assembly. For a plane slab the analysis, which is given in detail, is an extension of the theory of the anomalous skin effect, and the ineffectiveness concept plays the same role in elucidating the physical mechanism. In samples of other shapes, interaction with the field may not be so narrowly confined to a small fraction of effective electrons, but it is suggested that the diffusion of magnetic field nevertheless proceeds at a rate that is usually rather indifferent to shape.

Laboratory observation of elastic waves in solids
View Description Hide DescriptionCompressional, torsional, and bending waves in bars and plates can be studied with simple apparatus in the laboratory. Although compressional and torsional waves show little or no dispersion, bending waves propagate at a speed proportional to (f)^{1/2}. Reflections at boundaries lead to standing waves that determine the vibrational mode shapes and mode frequencies. Boundary conditions include free edges, simply supported edges, and clamped edges. Typical mode shapes and mode frequencies for rectangular bars, circular plates, and square plates are described.

Measurements of the superfluid transition in helium by means of a vibrating reed
View Description Hide DescriptionA relatively simple method is presented that has been used in an undergraduate experimental course for observing the superfluid transition in ^{4}He. The experimental technique consists of submerging a vibrating reed in liquid helium and observing the changes in frequency and damping (energy dissipation) as the temperature is regulated above and below the superfluid transition. The results of the measurements are interpreted by means of a simplification of the acting hydrodynamic forces, and it is seen that most of the relevant physics is well described by the approximations.

Conduction current and the magnetic field in a circular capacitor
View Description Hide DescriptionFrom the perspective of Ampere’s circuital law, either displacement current or conduction current can be viewed as the source for the magnetic field inside a circular capacitor that is slowly being charged. The Biot–Savart law is more selective. How it can be used with conduction current alone is shown. Also considered is the ‘‘leaky capacitor. Here it is shown that an isolated chargedcapacitor which discharges slowly in a homogeneous Ohmic dielectric produces no magnetic field anywhere. Alternatively, a field is produced if the conducting material is confined to a limited region. This field is calculated for a circular capacitor when only the material in the gap is conducting.

Coriolis and magnetic forces: The gyrocompass and magnetic compass as analogs
View Description Hide DescriptionThe Coriolis force on a particle (acting within a rotating frame) and the magnetic force on a charge, are both proportional to and act perpendicular to the velocity. As a consequence, the action of the Coriolis force on rotating mass, and the action of the magnetic force on rotating charge are formally identical. Just as the action of a magnetic field is to align the axis of the rotating charge distribution (magnetic dipole) with itself, so the Coriolis force aligns the axis of a rotating mass distribution (angular momentum) with the angular velocity of the rotating frame. This enables us to understand the gyrocompass by analogy with the magnetic compass. A simple way to demonstrate the gyrocompass to a class is also presented.

A hydrogenic atom in d‐dimensions
View Description Hide DescriptionThe laws of physics in dspatial dimensions are interesting and often lead to insights concerning the laws of physics in three spatial dimensions. A hydrogenic bound system in d‐dimensions is investigated where d is any real positive number, not necessarily an integer, and where a d‐dimensional radial Schrödinger equation is defined by the analytic continuation of the Schrödinger equation used in integer dimensions, with a d‐dimensional potential defined by Gauss’ law. The fundamental constants and physical relationships in d dimensions are arrived at by assuming the electric flux obeys a Gausslike law in d dimensions. This fixes the potential and makes it possible to write down the d‐dimensional Schrödinger equation. This equation is then solved numerically for the ground‐state energy eigenvalues, for several different dimensions, using a fourth‐order Runge–Kutta predictor algorithm. The solutions to this equation are compared to the analogous Bohr model energies in d dimensions and to the results obtained by others at well‐known integer dimensions. There is good qualitative agreement between the two theories in all dimensions. In addition, the wave functions for the Schrödinger equation are numerically determined and normalized in order to locate the most probable orbital radius. These values are compared to the d‐dimensional Bohr radii.

The two‐dimensional hydrogen atom with a logarithmic potential energy function
View Description Hide DescriptionRecently, a ‘‘shooting’’ method has been used to obtain exact expressions for eigenvalues and eigensolutions of the two‐dimensional hydrogen atom. This paper shows that the shooting method is easy for undergraduate students to understand and implement numerically. The highly accurate approximations for both eigenvalues and eigensolutions are then used to contrast the two‐dimensional and three‐dimensional hydrogen atoms. Finally, previous methods for solving the two‐dimensional hydrogen atom are compared with the shooting method.

Newton’s first two laws of motion are not definitions
View Description Hide DescriptionNewton’s first two laws of motion are often taken to be a definition of force. It is argued that they are true laws in that they make statements about the nature of the physical world. In particular, the first law can be viewed as asserting the existence of an ensemble of trajectories in a four‐dimensional space along which force‐free bodies, if they exist, would move. Such an ensemble, together with Newton’s absolute time, constitute the essential ingredients of the underlying geometrical structure of Newtonian space‐time. If this view is accepted, it is a relatively simple matter to describe how special and general relativity differ from Newtonian mechanics.

Precise calculation of the electrostatic force between charged spheres including induction effects
View Description Hide DescriptionThe method of images is applied iteratively to compute precisely the electrostatic force between charged spheres. The results are compared to the experimental results reported by Coulomb and show that Coulomb overlooked induction effects revealed in his data.
