Volume 44, Issue 3, March 1976
 Papers


Establishing an undergraduate research program in physics: How it was done
View Description Hide DescriptionIn the fall of 1969, MIT extended an open invitation to all of its undergraduate students to join the professional research activities of the faculty. This undergraduate research program rapidly increased in size, scope, impact, and diversity. In this article we report on the Undergraduate Research Opportunities Program (UROP) in the MIT Physics Department: the philosophy and objectives of the Physics program, its organization, rules, funding, growth, atmosphere, problems, and accomplishments. By the end of its first three years, Physics UROP included more than 50% of all the undergraduates enrolled in physics. Evaluations of UROP by faculty, administrators, and students were uniformly enthusiastic. This article presents the outline of how Physics UROP was established and how it evolved. Now in its seventh year, UROP has survived to maturity without becoming a cumbersome bureaucracy.

The rise and fall of PSI in physics at MIT
View Description Hide DescriptionSmall‐scale trials of the Personalized System of Instruction (PSI) in physics at MIT under the sponsorship of the Education Research Center were followed by adoption of large‐scale introductory PSI courses in the Department of Physics. After four semesters, all introductory physics PSI courses were suspended. This study investigates what happened. We point out organizational difficulties in going from small‐ to large‐scale use of the PSI method. We also identify certain conflicting perceptions among participants and observers that contributed to the suspension. The study, we hope, is an example of an informative ’’postmortem’’ of a type much needed in higher education.

Teaching general learning and problem‐solving skills
View Description Hide DescriptionThis 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.

Significant physics content and intellectual development—cognitive development as a result of interacting with physics content
View Description Hide DescriptionIncreasingly during the past 20 years intellectual development has been recognized as the central purpose of education. The Piagetian model of intellectual development tells us that each student must engage a subject in a manner appropriate to his or her p r e s e n t stage of development if he or she is to advance to the n e x t stage of development. When applied to college physics teaching, this theory implies that the large fraction of introductory physics students who are at the concrete operational stage of development must observe physical phenomena directly while they themselves are manipulating the equipment. Only in this way can they progress to the formal operational stage that characterizes professionals in the field.

Resource Letter RC‐1: Cosmology
View Description Hide DescriptionPrepared at the request of the AAPT Editorial Board for Resource Letters. This is one of a series of Resource Letters on different topics intended to guide college physicists,astronomers, and other scientists to some of the literature and other teaching aids that may help improve course contents in specified fields. No Resource Letter is meant to be exhaustive and complete; in time there may be more than one letter on some of the main subjects of interest. Comments on these materials as well as suggestions for future topics will be welcomed. Please send such communications to Professor Aaron Owens, Editor, Resource Letter Board, Department of Physics, Lake Forest College, Lake Forest, Illinois 60045.
N o t a t i o n: The letter E after an item number indicates elementary level; the letter I indicates intermediate level; and the letter A indicates advanced material. An asterisk (*) indicates those references we feel to be especially helpful.
R e p r i n t s: When ordering, request Resource Letter RC‐1. Enclose 75 cents per copy (not in stamps) together with a stamped and self‐addressed envelope and send to Melba Phillips, Acting Executive Officer, American Association of Physics Teachers,Graduate Physics Building, SUNY at Stony Brook, Stony Brook, New York 11794.

’’The Age of Newton’’: An intensive physics and mathematics course
View Description Hide Description’’The Age of Newton’’ is an intensive course in mathematics (calculus), mechanics, optics, and astronomy directed mainly toward nonscientists. Although it introduces the subjects in a consciously historical context, it is a course in—not merely about—science and mathematics. The course is divided into a central lecture series, at which guest lecturers may participate, a series of topical tutorial lectures, and four workshops or laboratories, one each in mathematics, mechanics, optics, and astronomy. Original sources, notably Newtons’s O p t i c s and Galileo’s T w o N e w S c i e n c e s, are used as far as practicable. Special apparatus was built to allow the performance of the experiments described by Newton and Galileo, in addition to modern experiments with air tracks, spectroscopes, lasers, and the rotating room.

’’Bohr and Einstein’’: A course for nonscience students
View Description Hide DescriptionA course that has been given at Michigan State University for nonscience upper‐class students is described. Its aim is to present the salient ideas of relativity and quantum physics in association with the lives of Albert Einstein and Niels Bohr. Generally qualitative arguments and discussions of key experiments are used to convey the scientific content. The controversy between Bohr and Einstein on quantum theory is included, and serves to point up a basic philosophical innovation in twentieth century physics. Some attention is also given to the attempts of both Einstein and Bohr to influence political events.

Implementing a course on the physics of music
View Description Hide DescriptionThis paper provides information to aid in starting a course in musical acoustics. The selection of topics for lecture, a project‐mode laboratory, texts, references, equipment, and lecture demonstrations are covered.

Observation of Rayleigh scattering and airglow
View Description Hide DescriptionA steerable 0.25‐m Ebert monochromator and photomultiplier detector were employed to measure the spectra of the clear daytime sky (Rayleigh scattering). The day sky intensity as a function of wavelength was observed to vary as λ^{−4} in agreement with Rayleigh’s predictions and observations. The same monochromator and detector with the addition of a multichannel scaler were used to observe the night sky spectra (airglow). The oxygen (OI) emission at 557.7 nm was observed from the ground by using this technique.

Connection between the dispersion relation, surfaces of constant energy, and energy bands in solids
View Description Hide DescriptionThe aim of this paper is to exhibit the relationship, for free electrons in two dimensions, between the dispersion relation ε=ε (k) = (h/^{2}/2m)(k _{ x } ^{2}+k _{ y } ^{2}), the contours of constant energy, and energy band structures in various directions. It has been found that these results, while well known, are not usually interconnected and presented together for the student.

A differential equation for the energy eigenvalues of relativistic hydrogenic atoms, and its solution
View Description Hide DescriptionBy using various general theorems, differential equations for the energy are written down for Klein–Gordon and Dirac hydrogenic atoms. When coupled with the fact that in the nonrelativistic limit one must get the Bohr formula, these equations are shown to determine the exact energy eigenvalues. Also, in the case of the Klein–Gordon equation, a recursion relation is derived from which the average values of powers of the radius can be found.

Refractive indices of gases at microwave frequencies
View Description Hide DescriptionA simple microwaveinterferometer capable of measuring small phase shifts is described. The interferometer can easily be incorporated in an undergraduate laboratory. The interferometer is used to determine the relative refractive indices of gases to within a few parts per million. Analysis of the reflection in the test chamber can provide an interesting extension to the usual treatment of multiple reflections.

Use of Doppler‐shifted light beams to measure time during acceleration
View Description Hide DescriptionIn this study of the twin paradox, the periods of acceleration necessary for the astronaut to start the journey, turn around, and return to rest at the starting point are investigated. Each twin sends a light beam of frequency f _{0} to the other one. The waves received by each twin have a constantly changing Doppler‐shifted frequency due to the astronaut’s acceleration. The stay‐at‐home twin (SAHT) receives K cycles sent by the astronaut and the astronaut receives K′ cycles sent by SAHT, where K≠K′. K and K′ are calculated by integrating the observed instantaneous frequency with respect to time. Then the time for the round trip recorded by the astronaut, T _{ A }, and by SAHT, T _{ S }, can be compared, where T _{ A }=K/f _{0} and T _{ S }=K′/f _{0}. The relationships derived are identical to those obtained by using the concept of proper time. The results are illustrated with numerical examples.

A simple model to explain the slowing down of light in a crystalline medium
View Description Hide DescriptionTwo main questions arise when one seeks to explain the propagation of light in a dielectric medium: (i) Why does the wave slow down? (ii) Why is there apparently only a forward wave? There is a subtle answer to these questions provided by the Ewald–Oseen extinction theorem, but the physical mechanism involved is not very easily visualized. This article sets forth a simple one‐dimensional model of parallel polarizable slabs that makes the mechanism clear and links the two questions in an interesting way.

One more derivation of the Lorentz transformation
View Description Hide DescriptionAfter 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 p r i n c i p l e 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 t h e o r y 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.

Probabilistic approach to teaching the principles of quantum mechanics
View Description Hide DescriptionA new approach to teaching the principles of quantum mechanics is presented that leads quickly to the Hilbert space postulates. After convincing the reader that an algebra of observables is a useful way to describe probabilistic observations, the paper introduces an evolution operator and demonstrates that it cannot commute with position observables. This fact, plus elementary assumptions about the possible form of the evolution operator, are sufficient to demonstrate that, if the deterministic limit of the theory is to yield Newton’s law, the evolution operator must have the familiar form of the Hamiltonian: essentially kinetic energy plus potential energy.

Analog computer demonstration of traveling and standing waves
View Description Hide DescriptionA simple analog computer program is described which, in conjunction with a dual‐trace storage oscilloscope, can demonstrate traveling and standing waves, the effect of various reflection coefficients, standing wave ratios, and other wave phenomena. The voltages generated by the analog are viewed on the oscilloscope screen to provide a vivid, easily understood, controllable demonstration, ideally suited for teaching.

Accurate orientation of the polar axis of an equatorial telescope mounting
View Description Hide DescriptionA straightforward method is given for determining the orientation of the polar axis of an equatorially mounted optical telescope. The background theory given illustrates some important properties of the celestial sphere and gives an insight into the usual ’’rule of thumb’’ methods of axis orientation; it also provides a basis for instructors and students to devise their own orientation methods.

Diffusion coefficient for a Brownian particle
View Description Hide DescriptionThis paper utilizes several existing techniques in nonequilibrium statistical mechanics (in a new context) to provide a relatively simple and short evaluation of the diffusion coefficient for a Brownian particle.

Natural units via linear algebra
View Description Hide DescriptionA systematic procedure is presented for introducing natural units, in which physical constants such as the speed of light in vacuum, Planck’s constant, and the mass of the electron are set equal to unity. The dimensions of physical quantities are represented by vectors in an abstract linear vector space, referred to as the d i m e n s i o n s p a c e. The quantities chosen as basic units play the role of basis vectors in this space. Conversion from one system of basic units to another corresponds to a different choice of basis vectors. Such a conversion is equivalent to a linear transformation on the dimension space and may be implemented by matrix multiplication. A detailed treatment is given of the transformation from mass–length–time units to a system of natural units appropriate to the atomic domain. The broader applicability of linear algebra techniques to dimensional analysis is briefly indicated by way of conclusion.
