Volume 60, Issue 3, March 1992
 Papers


Conformal transformations and the application of complex variables in mechanics and quantum mechanics
View Description Hide DescriptionThe use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self‐dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrödinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self‐dual in all dimensions. The methods may also be applied in the presence of magnetic fields.

Phase‐conjugate Michelson interferometers for all‐optical image processing and computing
View Description Hide DescriptionWhen the mirrors of a conventional Michelson interferometer are replaced by phase‐conjugate mirrors, the resulting interferometer displays dramatically altered behavior which makes it ideally suited to performing important operations in parallel optical image processing and optical computing. The phase‐reversing property and real‐time response of the phase‐conjugate mirrors make this type of interferometer much more immune to spatially uniform and nonuniform phase distortions, both static and dynamic. Phase‐conjugate interferometers are remarkably stable, sensitive only to r e a l a m p l i t u d e disturbances or information in the arms, and they readily display the subtraction, addition, intensity inversion, spatial differentiation and, in certain cases, temporal differentiation of two‐dimensional optical images.

A new angle on Compton scattering
View Description Hide DescriptionGenerations of physicists have been inculcated with the elementary Compton formula for the wavelength change of a photonscattered from a charged particle at rest (‘‘special’’ Compton effect). It does not seem to be appreciated that by allowing the target particle to move (‘‘general’’ Compton effect), and by redefining the photon’s scattering angle, one obtains a formula just as simple but more instructive. Unlike the standard formula, it directly displays: (a) number of degrees of freedom (for general effect), (b) time‐reversal invariance, (c) quadric surface of revolution—ellipsoid, etc.—for the locus of the scatteredphoton momenta, and (d) a criterion for occurrence of the inverse Compton effect (CE). It also: (e) facilitates interpretation of the CE as a double Doppler shift, and (f) extends the meaning of ‘‘Compton effect’’ to arbitrary targets—massive, massless, or with imaginary mass (tachyons). It reveals a connection between the Compton scattering of a tachyon in vacuum and its Cherenkov radiation.

Time evolutions of quantum mechanical states in a symmetric double‐well potential
View Description Hide DescriptionThe time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time‐dependent system is adequately described by a two‐state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency ν_{ t } is well approximated by the well‐known formula for the tunneling frequency ν_{ t }≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three‐state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three‐state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time‐dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double‐minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three‐state model varies slowly with energy.

Study of radiation‐matter interaction processes below 1 MeV from simulated data
View Description Hide DescriptionThe passage of 1‐MeV photons through thin and thick foils of Al and Pb has been simulated. The results have been used to show the basic properties of the dominant effects in radiation‐matter interaction processes at energies just below 1 MeV. For the interpretation of the data it is necessary to handle total and differential cross sections of the involved processes, as well as the conservation laws, mainly of the Compton effect, whose phenomenology is studied in depth. In particular, the effect of multiple interactions in thick foils is studied and some results of the simulation are compared with suggested theoretical approaches.

The pathological kinematics of unphysical force laws
View Description Hide DescriptionParticles subject to repulsive power‐law forces of position and propulsive power‐law forces of velocity exhibit dramatically pathological kinematics. Although the behaviors examined in this article are the result of the unphysical nature of these force laws and the surprising results might, as a result, be viewed as merely academic, understanding them is important for resolving the otherwise paradoxical results of some numerical simulations.

Conservation of linear and angular momentum and the interaction of a moving charge with a magnetic dipole
View Description Hide DescriptionThe operation of the laws of momentum and angular momentum conservation in the interactions between current‐carrying bodies and charged particles is analyzed using the correct expression for the force on a magnetic dipole, which takes into account the possible presence of hidden momentum in a current‐carrying body. At nonrelativistic velocities, Newton’s third law holds for the interactions, and thus the mechanical momentum associated with the motion of current‐carrying bodies and charged particles in a closed system is conserved itself in the nonrelativistic limit. There is no conflict with overall linear momentum conservation because the electromagnetic field momentum is equal and opposite to the hidden momentum of the current‐carrying bodies. However, the field angular momentum in a system is not compensated by hidden angular momentum, and thus only the sum of mechanical angular momentum, which must include any hidden angular momentum, and field angular momentum is conserved.

A laboratory course in computer interfacing and instrumentation
View Description Hide DescriptionThe rationale and content are described for an electronics‐oriented course in computer‐aided measurements. Performed on a prototyping board connected to an IBM PC‐AT, the experiments emphasize digital‐to‐analog and analog‐to‐digital conversion at the chip level. Techniques are presented for parallel and serial communication; waveform generation; the acquisition, display, and analysis of data; the use of graphics; control and feedback; mechanical positioning with stepper motors; and signal averaging. Students have undertaken a wide range of original projects.

Planck’s constant determination from black‐body radiation
View Description Hide DescriptionA method originally described by Crandall and Delord has been improved to measure the ratio of the Planck to the Boltzmann constant with low cost apparatus. Such a system can be easily implemented in high school or college laboratories. These improvements enable one to attain accuracies of 2% for Planck’s constant.

Shape invariance of Coulomb problems
View Description Hide DescriptionIt is shown that the Coulomb problems associated with the Schrödinger, Klein–Gordon, and Dirac equations are shape invariant. This property is used to obtain the energy eigenvalues and the normalized coordinate‐space eigenfunctions for bound states of these problems.

Optical properties of a metal film and its application as an infrared absorber and as a beam splitter
View Description Hide DescriptionThe optical properties of a thin metalfilm are derived, with a thin film defined as having a thickness smaller than the wavelength of the radiation and the skin depth. The derivation is based on the continuity equation for the electric field and on the energy balance for the incident, reflected, transmitted, and absorbed energy current. The use of thin metalfilms as anti‐interference coatings, as absorbers for the infrared, as semitransparent mirrors in interferometric devices and beam splitters, and as mirrors is discussed.

Displacement, velocity, and frames of reference: Phenomenographic studies of students’ understanding and some implications for teaching and assessment
View Description Hide DescriptionStudent understanding of fundamental concepts in kinematics has been explored using the phenomenographic research method. University and high schoolphysics students were interviewed and their understandings of displacement, velocity, and frames of reference have been analyzed in particular problem contexts. Descriptions of the different ways students understand the concepts have been developed and relations between the different levels of understanding have been identified. The data highlight the contextual nature of learning and the need for teachers to focus on the nature of student understanding in specific contexts using questions that require qualitative explanation by students. In particular, it is demonstrated that success in mechanical, quantitative problem solving can mask inadequate understanding of basic concepts that hinders learning in later years of study of the subject. Implications for teaching and assessment are discussed.

Phase measuring circuit with lead‐lag indication
View Description Hide DescriptionThis paper deals with the design and construction of a circuit for measuring the phase difference between two coherent signals in the audio frequency range. On application of two coherent signals to the inputs of the circuit an output voltage is produced that is proportional to the phase difference between the input signals. A novel feature of the circuit is that the polarity of the output voltage indicates which of the two signals leads in phase.
