Showing papers in "American Journal of Physics in 1976"

Optical theorem and beyond

Abstract: The history of the so‐called optical theorem in scattering theory is traced from its beginning over 100 years ago to recent applications of its generalizations.

One more derivation of the Lorentz transformation

Abstract: After a criticism of the emphasis put on the invariance of the speed of light in standard derivations of the Lorentz transformation, another approach to special relativity is proposed. It consists of an elementary version of general group‐theoretical arguments on the structure of space–time, and makes use only of simple mathematical techniques. The principle of relativity is first stated in general terms, leading to the idea of equivalent frames of reference connected through ’’inertial’’ transformations obeying a group law. The theory of relativity then is constructed by constraining the transformations through four successive hypotheses: homogeneity of space–time, isotropy of space–time, group structure, causality condition. Only the Lorentz transformations and their degenerate Galilean limit obey these constraints. The role and significance of each one of the hypotheses is stressed by exhibiting and discussing counterexamples, that is, transformations obeying all but one of these hypotheses.

Elementary derivation of the perturbation equations of celestial mechanics

Abstract: The equations of celestial mechanics that govern the temporal rates of change of orbital elements are completely derived using elementary dynamics and proceeding only from Newton's equation and its solution. Two orbital equations and the four most meaningful orbital elements - semimajor axis, eccentricity, inclination, and longitude of pericenter - are written in terms of the orbital energy (E) and angular momentum (H) per unit mass. The six resulting equations are differentiated with respect to time to see the effect on the orbital elements of small changes in E and H. The usual perturbation equations in terms of disturbing-force components are then derived by computing the manner in which perturbing forces change E and H. The results are applied in a qualitative discussion of the orbital evolution of particles in nonspherical gravitational fields, through atmospheres, and under the action of tides.

Teaching general learning and problem‐solving skills

Abstract: This article describes the investigation and teaching of two general cognitive skills important in introductory physics. We first analyzed the various abilities needed for understanding a relation (definition or law) well enough to use it appropriately. Then we developed two different instructional methods for teaching students the general learning skill of gaining such an understanding of any new relation. We further taught students a simple strategy for problem solving. Our results indicate that students can indeed be taught such general cognitive skills and that they can transfer these skills to areas outside of physics.

Singular potentials in one dimension

Abstract: In quantum mechanics of one dimension it is shown for potentials which become infinite at a point but are continuous elsewhere, that the singularity acts as an impenetrable barrier if the potential is not integrable up to the singularity, but if the potential is integrable the behavior is not essentially different from that of a potential which does not become infinite.

Multiple rainbows from single drops of water and other liquids

Abstract: The scattering of light by spherical drops and the theory of rainbows are reviewed in detail to predict the angular positions of rainbows from single drops. The angular positions of the first 13 rainbows of water, observed from a drop suspended in a student spectrometer, were compared to these theoretical positions. Since the rainbows were dependent on the refractive index, the positions shifted with drops of other fluids having other refractive indices. Light corn syrup yielded 11 rainbows, including the 17th‐order one. The light emerging from the drops was white except for the multicolors of the rainbows and the faint blue of the rays that had been nearly tangentially incident to the drops. With a hand‐held polarized filter, all of the observed rainbows appeared to be polarized parallel to the plane of incidence. Using the 6328‐A light from a student He–Ne laser, one of the interference patterns composing all natural rainbows was created and photographed. The angular positions of the principal maxima of this pattern differed from the theoretical rainbow positions predicted by geometrical optics by amounts dependent on the drop radius.

Cultivating the capacity for formal reasoning: Objectives and procedures in an introductory physical science course

Abstract: Experience in a physical science course for preservice elementary teachers, inservice elementary teachers, and other nonscience majors has led to the identification of certain specific factors and procedures that seem to assist the attainment of the formal operational level of intellectual development on the part of students who are initially at a concrete or transitional stage.

Spin and statistics

Abstract: An exchange operator is defined in terms of rotation and displacement operators This operator contains a rotation around the z axis that produces a factor (−1)2s in front of the wave function By postulating that such a rotation must produce the same value for the wave function if the particles are identical, we obtain the usual connection between spin and statistics Particle states lying in the irreducible representations of the three‐dimensional rotation group are seen to allow only Fermi and Bose statistics

On the generally covariant Dirac equation

Abstract: Using explicit tetrads for any given metric and Fock–Ivanenko coefficients of the generally covariant Dirac equation, we develop a formal method to relate the wave function in a noninertial frame to that in the inertial frame The resulting equivalence transformations are exact The familiar cases of the rotating frame and generalized hyperbolic motion are given as examples

Maxwell form of the linear theory of gravitation

Abstract: A treatment of the linear theory is given that closely parallels the usual E and B formulation of electromagnetic theory. In Part I, an introductory exposition of gravitational radiation is presented. The tidal gravitational field strength E is defined, and it is shown that E is the measurable quantity which characterizes a gravitational wave. The important point is made that the problem of detecting gravitational waves is really the same problem as detecting tidal accelerations due to a time‐varying Newtonian gravitational field. In Part II, the magnetic‐type gravitational field strength B is defined and Maxwell‐like field equations are used to determine the polarization states of a gravitational wave. The detection of gravitational waves by Weber‐type cylinders is discussed.

Theoretical and Experimental Study of the Motion of the Simple Pendulum.

Abstract: A pedagogic experimental and theoretical study of the motion of the simple pendulum, which considers corrections to the approximation of simple harmonic motion, is presented. The period is calculated with the use of both elliptic integrals and the perturbation theory of Kryloff and Bogoliuboff. Measurements of the period of the pendulum taken with a stopwatch are compared with the calculations. The errors made in the measurements are analyzed.

Personalized instruction: A summary of comparative research, 1967–1974

Abstract: This paper summarizes the results of fourteen separate studies comparing the learning outcomes of a new instructional procedure, the ’’Personalized System of Instruction’’ (PSI), with the learning outcomes of conventional approaches to college teaching. The major conclusion suggested by this summary of research is that, when evaluated by average student performance on course content examinations, PSI has proven superior to the conventional methods with which it has experimentally been compared. The author examines features of PSI courses which probably account for this superiority, and he discusses some of the theoretical and practical implications of research on PSI conducted between 1967 and 1974.

Fast Fourier transform: An introduction with some minicomputer experiments

Abstract: The Cooley–Tukey fast Fourier transform (FFT) has had an extraordinary impact on the computation of Fourier transforms. A tutorial account is given of how the algorithm works and of its relationship to the more familiar continuous Fourier transform and Fourier series. Some pitfalls associated with sampled data over a finite window are outlined. Several examples are given illustrating how the FFT is useful as a teaching tool to introduce the subtleties of spectral analysis of sampled data by interactive minicomputer experiments. A bibliography to the literature is given.

Diffraction in time and the time–energy uncertainty relation

Abstract: The purpose of this paper is to discuss two forms in which the time–energy uncertainty relation can appear. One of them concerns the time of preparation of a given state and the other the lifetime of compound systems. Both will be illustrated here through exactly solvable examples. The first one will be the opening and subsequent closing of a shutter on which we have an incident beam of particles. This problem shows a diffraction‐in‐time effect closely connected with the first type of time–energy uncertainty relation. The second will be the decay of a compound state described through a schematic R matrix formalism. There one can clearly see the transition between the time–energy uncertainty relation as associated with the interval in which the decay process has been in operation, and as associated with the lifetime of the compound state.

Obtaining a Heisenberg Hamiltonian from the Hubbard model

Abstract: A pedagogically useful method of obtaining a Heisenberg spin–spin interaction from a familiar many‐electron Hamiltonian by means of ordinary degenerate perturbation theory is presented.

Physics teaching: Does it hinder intellectual development?

Abstract: A necessary condition for effectively teaching physics is that the students have the capability of operating at the cognitive level that is matched with the logical structure that produced the discipline. Recent research indicates that a majority of students are not demonstrating this capacity. The analysis of the performance of students ’’learning’’ physics without the necessary cognitive development demonstrates behavior that hinders their intellectual growth.

De Broglie’s thesis: A critical retrospective

Abstract: Louis de Broglie’s doctoral thesis developed a concept of waves associated with material particles that was soon incorporated into wave mechanics and later supported by experimental demonstrations. De Broglie’s original development, however, relied on an incorrect identification of two quite different relations: the relation between the velocity of a particle and the relation between the group velocity of a wave packet and the velocity of individual waves in the packet. A clarification of this difference and its historial significance is followed by a tentative account of the reasons why de Broglie’s well‐known formula proved successful, though the theory supporting it rested on a conceptual confusion. Finally, this development and clarification are related to some of the difficulties involved in the attempts top fashion a coherent interpretation of quantum theory.

How children swing

Abstract: Everyone knows how to do it, but not everyone knows how it works. In this analysis of the motion of a playground swing, it is shown that, under conditions of constant pumping, the stored energy increases approximately exponentially with time. A comparison is made between this system and other parameteric amplifiers in various fields of physics, and the problem of starting a swing from rest is discussed.

Why does a mass on a spring sometimes misbehave

Abstract: Autoparametric resonance of a mass on a spring was investigated It is shown that when a spring is loaded so that it stretches to four-thirds of its original rest length, the spring frequency will be double the pendulum frequency and it will not be possible to excite a pure verticcal spring oscillation (GHT)

Phase‐space of a driven, damped pendulum (Josephson weak link)

Abstract: We have generated a number of phase‐space diagrams for several physically significant differential equations, including those describing the motion of a driven, damped pendulum and a Josephson weak link. These phase‐space diagrams can lead to considerable insight into the complex behavior exhibited by these systems. A number of these phase‐space diagrams were generated with parameters that can be duplicated with suitable mechanical models, allowing them to be used in conjunction with such models for a detailed study of the behavior of the driven, damped pendulum and Josephson weak link.

Bound‐state wave packets

Abstract: We point out that a convenient way to construct wave packets is to superimpose bound‐state eigenfunctions. The resulting packets can be used to study spreading, scattering, and reflection as well as features unique to bound‐state phenomena. The infinite square well and the infinite square well with a δ‐function barrier are illustrated.

Comment on ’’Negative Kelvin temperatures: Some anomalies and a speculation’’

Abstract: ln this note, we prove that for systems that can exhibit in the space of thermodynamic variables we are looking for negative Kelvin temperature there exist no adiabatic surthe existence of a path going anywhere through the hyfaces connecting regions of opposite temperature sign. perplane p = O and such that the entropy is constant on that The motivation for this proof comes from a note pubpath. We shall thus prove that the value of S in the hyperlished a few months ago by Tykodiin this Journal,' in which plane 0 = O can only be different from its value in neighhe recalled certain properties of systems exhibiting negative bouring hyperplanes, and then Pippard's statement will be Kelvin temperatures and pointed out that it might be necproved. More specifically, we claim the following: essary to formulate a new law of thermodynamics to forbid the following processes: (a) the running of a Carnot \"cycle\" between a reservoir of finite temperature and one of infinite temperature since it would permit a reversible 100% conversion of heat into work (it is known that at negative temperature the Kelvin-Planck formulation of the second law must be modified to permit irreversible 100% conversion of heat into work2); (b) the existence of Carnot cycles working between reservoirs of temperatures of opposite signs, since such cycles would perform work and at the same time would \"pump\" entropy from the colder to the hotter reservoir. Pippard3 mentions that \"no isentropic surfaces connect positive and negative temperatures.\" It is clear that no new law would be needed if one could prove such a s ta te ment.^ We shall present such a proof here. It involves nothing more than a plausible hypothesis about systems that can exhibit negative temperature, and it uses very elementary relations of statistical mechanics that can be found in any of the classical texts on the ~ u b j e c t . ~ In the canonical ensemble, the entropy is written5

The beginning of quantum statistics: A translation of ’’Planck’s law and the light quantum hypothesis’’

Abstract: The phase space of a light quantum in a given volume is subdivided into ’’cells’’ of magnitude h3. The number of possible distributions of the light quanta of a macroscopically defined radiation over these cells gives the entropy and with it all thermodynamic properties of the radiation.

Goethe’s theory of color and scientific intuition

Abstract: A summary of Johann Wolfgang von Goethe’s color studies is presented with special attention paid to his ’’method.’’ It is proposed that the act of accurate qualitative observation creates the capability in the observer for an intuitive understanding of the physical laws underlying the phenomena under observation. The use of such a method as a basis for laboratory instruction is discussed.

Teacher education and the implementation of elementary science curricula

Abstract: A program is described to prepare in‐service elementary school teachers in the subject matter and reasoning skills needed to teach science as a process of inquiry. Included is a procedure for introducing experience‐oriented curricula into the schools. Selected teachers are prepared in a university physics department to serve, under close supervision, as instructors for their colleagues. The program has yielded results with implications for what physics departments should provide for elementary school teachers. Intellectual difficulties encountered by the teachers are discussed that have a bearing not only on successful implementation of school science programs but on physics education in general.

Electromagnetic Fields in Rotating Superconductors.

Abstract: Electric and magnetic fields induced inside rotating superconductors are explained in terms of the familiar centrifugal and Coriolis forces.