Volume 61, Issue 6, June 1993
 Editorial
 Papers


An introduction to geometric algebra with an application in rigid body mechanics
View Description Hide DescriptionThis paper presents a tutorial of geometric algebra, a very useful but generally unappreciated extension of vector algebra. The emphasis is on physical interpretation of the algebra and motives for developing this extension, and not on mathematical rigor. The description of rotations is developed and compared with descriptions using vector and matrix algebra. The use of geometric algebra in physics is illustrated by solving an elementary problem in classical mechanics, the motion of a freely spinning axially symmetric rigid body.

An introduction to geometric calculus and its application to electrodynamics
View Description Hide DescriptionA tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.

More than one mystery: Quantum interference with correlated charged particles and magnetic fields
View Description Hide DescriptionExamples of charged‐particleinterferometry are descibed whereby isolated magnetic fields—i.e., fields through which the particles do not pass—influence the clustering, as well as spatial distribution, of the particles. The configurations, which combine features characteristic of (i) Aharonov–Bohm and Hanbury Brown–Twiss experiments, and (ii) Aharonov–Bohm and Einstein–Podolsky–Rosen experiments, manifest simultaneously the effects of three distinct kinds of interference: (i) interference, dependent upon optical path length difference, resulting from wavelike propagation of particles; (ii) interference, dependent upon confined magnetic flux, resulting from particle charge and spatial topology; and (iii) interference, dependent upon quantum statistics, resulting from particle indistinguishability under exchange.

Kinetic theory: Understanding nature through collisions
View Description Hide DescriptionThe attempts to understand the macroscopic behavior of fluids, in particular their nonequilibrium properties, in terms of the dynamics of the many particles, out of which they consist, are sketched from Clausius, in the middle of the 19th century, to the present day. The fundamental role that collisions play for dilute as well as dense gaseous or liquid systems is emphasized. The main difficulty is to identify the relevant collision sequences in the myriads of collisions that go on in a macroscopic system. While for dilute gases uncorrelated binary collisions dominate, as was already known in the 19th century, relevant correlated collision sequences for dense gases and liquids have only been identified in the last 25 years, leading to vortex and cage diffusion. Applications to transport coefficients are discussed.

Light polarization: A geometric‐algebra approach
View Description Hide DescriptionThe geometric algebra of three‐dimensional space (the ‘‘Pauli algebra’’) is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three‐dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density ρ is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three‐dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of ρ is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of ρ. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.

Dipole radiators in a cavity: A radio frequency analog for the modification of atomic spontaneous emission rates between mirrors
View Description Hide DescriptionThe results of a simple experiment that studies the dipole radiation rate of a radio frequency antenna inside a parallel mirrorcavity are presented. Data on the emission rate of the dipole as a function of both mirror spacing and the dipole location between mirrors is taken, and agrees well with a theoretical prediction that holds in both the classical macroscopic and quantum electrodynamical microscopic domains: In particular, theoretically predicted emission suppression and enhancements are observed. This project is then of pedagogical interest in that it is a simple mockup of several famous cavity QED experiments that investigate the change of atomic spontaneous emission rates in such resonators.

Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession
View Description Hide DescriptionThe frequency of Thomas precession and the magnitude of spin–orbit coupling are shown to follow directly from conservation laws of momentum, angular momentum, and energy in relativistic classical mechanics, without transforming to axes that are instantaneously at rest with respect to the orbiting spinning particle.

Parametric solution of the phase diagram of a mean‐field Ising model exhibiting a tricritical point
View Description Hide DescriptionA mean‐field approach to an Ising lattice with two‐spin (pair) and four‐spin (quartet) interactions is presented as an instructional example of a system exhibiting a tricritical point. In the absence of an external magnetic field, it is shown that the phase diagram of this system presents second‐ and first‐order transition lines in accordance with the relative strength of the pair and quartet interaction constants. These lines meet at a tricritical point. A parametric solution of the metastability (spinodal) and coexistence curves is reported.

Experimental verification of the Heisenberg uncertainty principle—An advanced undergraduate laboratory
View Description Hide DescriptionThe Heisenberg uncertainty principle can be experimentally demonstrated by combining a Mössbauer experiment with a measurement of a nuclear lifetime. The senior undergraduate students perform a Mössbauer experiment to measure the energy width of the 14.4 keV level of ^{57}Fe followed by measurements of coincident γ rays to determine the lifetime of the level. The experiments are designed to emphasize that the uncertainty principle is inherent in the wave function rather than resulting from the measurement process.

Transmission spectroscopy of a thin membrane
View Description Hide DescriptionA few‐micron‐thick nitrocellulose membrane stretched flat across an open frame (a ‘‘pellicle’’) has gently modulated passbands typically spaced by 20–60 nm across the visible and near‐infrared. This makes it an attractive subject for an upper‐division laboratory exercise. In our experiment, students determine the membrane’s thickness d and refractive indexn from spectra recorded over a range of incidence angles. These spectra illustrate the classical textbook problem of transmission by a parallel plate and yield values of d and n with uncertainties less than 1%.

Cooling of a vertical cylinder by natural convection: An undergraduate experiment
View Description Hide DescriptionThe transient cooling of a vertical cylinder by natural convection is studied in a computerized undergraduatelaboratory experiment. The dependence of the heat transfer coefficient on temperature is determined using simple data fitting algorithms. The results of the experiment clearly show that the natural convectionheat transfer coefficient is proportional to ΔT ^{1/4}, in agreement with previously published values.
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 Book Reviews


Atom–Photon Interactions: Basic Processes and Applications
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POST‐USE REVIEW. Physics, 3rd Edition
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Introduction to MathCAD for Scientists and Engineers
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Great Ideas in Physics
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