Volume 50, Issue 6, June 1982
 Letters To The Editor



Archimedes’ principle
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Alternative choice for the energy flow vector of the electromagnetic field
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 Editorial


Editorial: Education for the future. I
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 Papers


Resource Letter GI‐1: Gravity and inertia
View Description Hide DescriptionThis Resource Letter provides a guide to the literature on gravity and inertia. The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A indicates rather specialized or advanced material. An asterisk (*) indicates articles that we feel are especially useful or interesting; a double asterisk (**) indicates those articles to be included in an accompanying reprint book.

Even honors students have conceptual difficulties with physics
View Description Hide DescriptionHonors students in an introductory physics course are shown to exhibit some of the same kinds of misconceptions as do students in the usual standard introductory courses. Examples are given of exercises and written exam questions that probe for conceptual understanding, and student responses to these questions are used to identify conceptual difficulties common to many students. Because these misconceptions were found in a very select group of students, the implication may be drawn that conceptual difficulties of the same kind are present in students in all levels of introductory physics.

Heuristic extension of the Schwarzschild metric
View Description Hide DescriptionThe Schwarzschild solution of Einstein’s equations of gravitation has several singularities. It is known that the singularity at r = 2G m/c ^{2} is only apparent, a result of the coordinates in which the solution was found. Paradoxical results occuring near the singularity show the system of coordinates is incomplete. We introduce a simple, two‐dimensional metric with an apparent singularity that makes it incomplete. By a straightforward, heuristic procedure we extend and complete this simple metric. We then use the same procedure to give a heuristic derivation of the Kruskal system of coordinates, which is known to extend the Schwarzschild manifold past its apparent singularity and produce a complete manifold.

Elementary derivation of the magnetic flux quantum
View Description Hide DescriptionStarting with the well‐known de Broglie relation m v = h/λ that holds for a particle in zero magnetic field we give an elementary derivation of the generalized de Broglie relation that holds for a charged particle in a circular orbit in a cylindrically symmetric magnetic field. We make no use of ’’div, grad, curl, and all that’’ and do not introduce canonical momentum, the vector potential, or the Schrödinger equation. This generalized de Broglie relation is then assumed to hold for each of the Cooper pairs in a superconducting hollow cylinder. With the further assumption of the Meissner effect we find the well‐known result φ = n h/q for the flux φ trapped by the circulating Cooper pairs of charge q = 2e. We then use the generalized de Broglie relation to show that the Cooper pairs have velocities about 10^{−6} times too small for them to be in equilibrium ’’cyclotron orbits’’ in the magnetic field they experience. We also show that this de Broglie relation gives the correct value (i.e., the Schrödinger theory value) for the London penetration distance.

Mirage in the laboratory
View Description Hide DescriptionThe theory of the mirage is reviewed at several levels of complexity, in order to show that the phenomenon can usefully be discussed in teaching. In addition, an experimental device has been set up to obtain the mirage in the laboratory under controlled conditions, with the possibility of observing the finer details discussed in the theory.

Iterative solutions of transcendental equations of mathematical physics with the programmable pocket calculator
View Description Hide DescriptionA method is outlined that permits the solution of transcendental equations, such as often arise in mathematical physics, using short iterative routines ideally suited to the programmable pocket calculator. The method is outlined, conditions for convergence are discussed, and examples are given.

Gravitational deflection of fast particles and of light
View Description Hide DescriptionThe gravitational deflection of relativistic particles and of light is discussed from a simple point of view, in terms of velocity‐dependent forces in flat space. This picture is shown to be equivalent to that of general relativity, and some of its metric extensions, in the limit of weak gravitational effects. The deflection can then be predicted, up to one free parameter. The relation of that parameter to the spin of gravitational radiation is considered, giving a simple discussion of the known connection between general relativity and a theory with spin‐two fields.

Can one measure the one‐way velocity of light?
View Description Hide DescriptionAs Fung and Hsieh maintain, one can readily measure the speed of light in one direction if it is different from c/(1−α cos ϑ) 0?α?1. On the other hand, if the velocity of light is c/(1−α cos ϑ), the properties observed upon reflection from a plane mirror do not allow one to conclude α = 0 (as they claim) but rather that α cannot be close to unity. Finally, it is shown that if the one‐way velocity is c/(1−α cos ϑ), a wide variety of experiments fail to yield a value for α.

Another look at the uniform rope sliding over the edge of a smooth table
View Description Hide DescriptionThe motion of a uniform rope sliding over the edge of a table has been repeatedly used to exemplify systems with variable mass. This usually involves a division of the system into two parts, which are considered separately. As a result, some physical information is lost and the motion of the rope looks somewhat awkward. The problem is re‐examined by properly considering the rope as a fixed‐mass two‐dimensional system. Several questions, including under what conditions will the rope keep the ’’inverted L shape,’’ are also studied.

Continuum eigenfunction expansions and resonances: A simple model
View Description Hide DescriptionA one‐dimensional square‐well model of an atom in an electric field is solved exactly. This problem is instructive for two reasons: (1) The Hamiltonian has a continuous spectrum, but the eigenfunctions do not reduce asymptotically to plane waves. Therefore the general Titchmarsh–Kodaira theory of eigenfunction expansions is needed to normalize the eigenfunctions properly. The normalization is expressed by a spectral density function, ρ(E). (2) ρ(E) exhibits ’’bumps’’ at values of the energy at which the electron wave function’s amplitude inside the well is particularly large. These are resonant states of the system. The gradual sharpening of the resonances into discrete bound states as the electric field is turned off is demonstrated.

Zitterbewegung and the Klein paradox for spin‐zero particles
View Description Hide DescriptionThe Hamiltonian form of the Klein–Gordon equation is used to study the Z i t t e r b e w e g u n g of spin‐ zero particles. It is shown that the time dependence of the velocity and position operators in the Heisenberg picture is formally the same as that of a spin‐ 1/2 particle. It is also shown that the Z i t t e r b e w e g u n g is the result of the interference between positive and negative energy states. A representation‐free expression is found for the mean position operator. This operator is free from Z i t t e r b e w e g u n g. The Klein–Gordon equation is solved with an electrostatic potential originally used by Sauter in his investigation of the Klein paradox for spin‐ 1/2 particles. It is shown that the transmission to states with negative kinetic energy is very small when the electric field is weak.

Quark confining potential in relativistic equations
View Description Hide DescriptionA linear potential, when used in the Schrödinger equation, confines a quark. In this paper, we discuss what happens when this potential is used in the relativistic Klein–Gordon and Dirac equations.
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 Notes and Discussions


Discontinuity in the first derivative of the Schrödinger wave function
View Description Hide DescriptionBound state problems in quantum mechanics are considered for the edge of a square well potential and singular points of a Coulomb and a δ‐function potential. (AIP)

Liquid lens
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Greninger nets for backscattered Laue analysis
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Evaluation of some averages for the hydrogen atom
View Description Hide DescriptionThe Hellman‐Feynman theorem is used to calculate 〈r ^{−1}〉, 〈r ^{−2}〉, and 〈r ^{−3}〉 for the hydrogen atom. (AIP)
