Volume 57, Issue 4, April 1989
 Guest Comment
 Editorial


Editorial: AJP Reviewers—Thank You!
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 Papers


American Association of Physics Teachers: Citations for Distinguished Service for 1989
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Molecular beam experiments, the Lamb shift, and the relation between experiments and theory
View Description Hide DescriptionThe Rabi molecular beamexperiments and the Lamb shift experiment are analyzed to elucidate the role the nonrelativistic Schrödinger equation plays in their design, analysis, and interpretation.

How big is a cyclotron?
View Description Hide DescriptionThe radius of the orbit of a particle of given energy in a laboratory cyclotron is computed in terms of the Bohr radius of the hydrogen atom a _{0} and the fine structure constant α.

Geometrical interpretation of the simultaneous diagonalization of two quadratic forms
View Description Hide DescriptionThe theory of small oscillations, when cast in matrix form, leads to an eigenvalue problem involving the simultaneous diagonalization of two real symmetric matrices. While a geometrical interpretation of this process exists, its visualization is generally hampered by the multidimensional nature of the coordinate transformations involved. I consider a model problem with just two degrees of freedom for which the coordinate transformations become planar and therefore easily visualizable. A set of plane diagrams is provided showing how the diagonalization is achieved for the model problem considered.

Fresnel lenses
View Description Hide DescriptionThis article contains a description of the discovery of the stepped lenses that today bear Fresnel’s name, based on a monograph written by Fresnel himself in 1822. Although it stands in the shadow of the rest of his activity, in particular the theory of transverse light waves, including polarization,diffraction, and birefringence, this invention could be regarded as a very interesting and very early result of government‐sponsored research activity in science and technology, to be used in lighthouses for at least a century, and to remain important in other fields of modern optics.

Introducing Schrödinger’s cat in the laboratory
View Description Hide DescriptionA laboratory session that is designed to introduce students to the conceptual foundations of quantum mechanics is described. In this laboratory session, several experiments are used in order to illustrate the important ideas of superposition, interference, and the ‘‘collapse’’ of the wavefunction. These experiments form the basis for a discussion of four of the most popular interpretations of quantum mechanics that have been proposed over the years: the instrumentalist interpretation, hidden variable theories, the objective view of the wavefunction, and the many‐worlds interpretation. As a finale, students are introduced to the paradox that epitomizes the interpretational problems of quantum mechanics: the paradox of Schrödinger’s cat.

Timely equations of rocket motion and the surprising power of rockets
View Description Hide DescriptionCurrent general physicstextbooks derive equations for rocket acceleration and rocket velocity as functions of mass. Rocket position is ignored in nearly all textbooks. Using time left till total mass consumption as a variable, simple equations for acceleration, velocity, and position can be easily derived. The flight of a rocket subject to constant gravitational acceleration proves quite simple to analyze. Finally, a power analysis shows that in an inertial reference frame, the power of a rocket can exceed the power of its engine.

A simple method for measurement of the diffusion of vapors
View Description Hide DescriptionA simple method for the measurement of the diffusion of vapors is described. Commercially available microcapillary tubes, a ruler, and a watch, allow measurement of the diffusion constant to a precision of better than 1%. This article describes measurements of the diffusion of water, benzene, and various straight chain alcohol molecules in air. The technique is suitable for the undergraduate laboratory or as a home experiment.

Galilean transformation and the path integral propagator for a crossed electric and magnetic field
View Description Hide DescriptionA Galilean transformation is made in the path integral formalism to obtain the Feynman propagator of a charged particle in the presence of a crossed uniform electromagnetic field in terms of the propagator of a charged particle in the presence of only a magnetic one. Also noted is how the result makes it possible to relate the initial propagator, with E and B, to that of the free particle.

Time‐reversal invariance in multiple collisions between coupled masses
View Description Hide DescriptionThe time evolution of two mechanical oscillators coupled by a spring can (but need not) exhibit an instant t=2t’ when the initial conditions at t=0 have been exactly restored. When that is the case, then at t=t’ energy and momentum have been exchanged exactly as in an elastic collision between two free particles, and the evolution of the system from t=t’ to 2t’ is related to that from 0 to t’ by time‐reversal invariance. A similar ‘‘simulation of elasticscattering’’ at t=t’ can occur for two free particles coupled via collisions with an intermediary mass that bounces back and forth between the two particles provided the intermediary is left at rest at t=t’. Examined here is the time evolution of the exchange of momentum and energy for these two examples, determining the values of the coupling spring constant (or mass value) of the intermediating spring (or mass) needed to simulate single elasticscattering between free particles, and looking at the manifestation of time‐reversal invariance.

An approximate analytic solution of the spectral inverse problem
View Description Hide DescriptionAn important problem of nonrelativistic quantum theory is: G i v e n a bound‐state energy spectrum, determine the space dependence of a potential function that will produce that spectrum. This is sometimes called the ‘‘spectral inverse’’ problem. Here, by use of the first‐order WKB energy quantization condition, an a p p r o x i m a t e analytic solution of this problem is given for the case of one‐dimensional symmetric oscillators and for a class of radial oscillators. This is done by reformulating the WKB quantization condition as an Abel integral equation that relates the quantum number to the potential’s space dependence. The resultant integral equation can be solved exactly. The harmonic oscillator and the hydrogen atom problems are used to demonstrate the method. The solution for the case of one‐dimensional asymmetric oscillators is also briefly discussed and the Morse oscillator is used as an example. Finally, the history of the spectral inverse problem is reviewed and the conditions necessary for its exact solution are discussed. In particular, the conditions required for the inversion process to be unique are considered.

Deriving a formula for nuclear radii using the measured atomic masses of elements
View Description Hide DescriptionMost nuclei have a nearly spherical shape and can be characterized by an effective radius R=R _{0} A ^{1} ^{/} ^{3}, where A is the nucleon number and R _{0} is a constant to be determined from experimental data. Traditionally, the data came from experiments on muonic or pionic x‐ray emission, high‐energy electron scattering, optical or x‐ray isotope shifts, β emission by mirror nuclei, α decay, or neutron scattering. Because this type of information is not readily available in textbooks or the pedagogical literature, it is difficult to ask students to establish the radius formula on their own. However, as shown here, one may easily circumvent the difficulty by using the readily available and highly accurate masses of the mirror nuclei. Results are presented for all existing pairs of mirror nuclei.

Heuristic methods for counting and sizing of molecules in the liquid state
View Description Hide DescriptionThough the liquid state is highly complex, there are global classical properties that liquids enjoy. Heuristic liquid‐state relations may be derived from the fundamental physics of surface energy and volume energy. It has long been known that the dimensional difference between these two measures can be exploited to yield estimates of molecular count and size. Herein is described an alternative approach in which a key parameter is the temperature coefficient of surface tension. This parameter circumvents the need for awkward volume energy measurements. Here indicated is how simple experiments, or even pretabulated chemical data, may be used to estimate such quantities as Avogadro’s number.

A classroom exercise to determine the Earth–Moon distance
View Description Hide DescriptionA simple indoor exercise to obtain the value of the Earth–Moon distance, in terms of the Earth’s radius, is described. The student measures on a photograph the ratio of the diameter of the Moon to the diameter of the shadow of the Earth during a lunareclipse. A simple but adequate reduction of the measurement is given.

Redesigning courses and textbooks for the twenty‐first century
View Description Hide DescriptionThere is evidence that a confluence of pressures to change the current physics syllabus is building from many sources. Texts for science and engineering students do not usually have much contemporary physics content. There is a need to update textbooks to include areas of current interest in physics research, to include important facets of 20th‐century physics that have gone virtually unnoticed in the present generation of physicstextbooks. The consequence of additions of topics must be restructuring of the physics curriculum and may lead to corresponding deletions among topics presently discussed.

A remark on the gravitational field produced by an infinite straight string
View Description Hide DescriptionThe results predicted by Newtonian gravity and general relativity are compared regarding the field produced by an infinite gauge string with constant density λ. A simple gedankenexperiment is suggested to stress the remarkable differences between these two theories. The existence of the usual Newtonian limit is discussed in this case.

Exercises in the synthesis of electrical impedances
View Description Hide DescriptionTwo general circuits are developed for the synthesis of user‐variable electrical impedances. One circuit permits user‐variable enlargement of any existing impedance without affecting its phase characteristics. This circuit facilitates such devices as floating capacitors and inductors which are continuously variable over wide ranges. The other circuit permits simulation of user‐variable floating impedances whose phase and frequency characteristics are widely subject to design objectives. In addition to all conventional impedance functions, it poses access to an infinite set of alternatives for the simulation of resistance and reactance functions of frequency. Included is a theoretical approach for predicting parasitic impedances reported in earlier works on similar devices known as gyrators.

Measurement of refractive index by double‐exposure speckle pattern recording
View Description Hide DescriptionAn experimental approach is presented to the measurement of the refractive index of materials based on the optical properties of speckle patterns. The sample being characterized is shaped in the form of a plane parallel plate and so inserted in a speckle‐forming beam. Speckles are recorded on a photographic plate with a double‐exposure technique producing a twin speckle pattern. Decoding of such a recorded pattern leads to the desired information.
