Volume 72, Issue 3, March 2004
 AWARDS
 PAPERS


Heisenberg in Poland
View Description Hide DescriptionWhile much is known about Heisenberg’s visit to Copenhagen in 1941 little has been written about his visit to Poland two years later. This article attempts to fill in the gap.

Physics of overarm throwing
View Description Hide DescriptionMeasurements are presented of the speed at which objects of different mass can be projected by an overarm throw. Light objects can be thrown faster than heavy objects, although the difference in speed is not as large as one might expect. For a factor of 60 increase in the thrown mass, there was a decrease of only 2.4 in the throw speed. The relatively small change in throw speed is due to the fact that the force that can be applied to a thrown object increases with object mass. Estimates of the muscle forces involved indicate that the increase in force with mass is primarily an inertial rather than a physiological effect. The total kinetic energy of the mass, hand, and the forearm was found to be almost independent of the object mass, and the throw speed is almost independent of the mass of the upper arm.

The evolution of radiation toward thermal equilibrium: A soluble model that illustrates the foundations of statistical mechanics
View Description Hide DescriptionIn 1916 Einstein introduced the first rules for a quantum theory of electromagnetic radiation and applied them to a model of matter in thermal equilibrium with radiation to derive Planck’s blackbody formula. Einstein’s treatment is extended here to timedependent stochastic variables, which leads to a master equation for the probability distribution that describes the irreversible approach of his model to thermal equilibrium and elucidates aspects of the foundations of statistical mechanics. An analytic solution of the master equation is obtained in the Fokker–Planck approximation, which is in excellent agreement with numerical results. It is shown that the equilibrium probability distribution is proportional to the total number of microstates for a given configuration, in accordance with Boltzmann’s fundamental postulate of equal a priori probabilities. Although the counting of these configurations depends on the particle statistics, the corresponding probability is determined here by the dynamics which are embodied in Einstein’s quantum transition probabilities for the emission and absorption of radiation. In a special limit, it is shown that the photons in Einstein’s model can act as a thermal bath for the evolution of the atoms toward the canonical equilibrium distribution. In this limit, the present model is mathematically equivalent to an extended version of the Ehrenfests’s “dogflea” model.

Putting your heart into physics
View Description Hide DescriptionWe describe techniques for measuring the time interval between successive heartbeats. This time series data can be used in undergraduatephysics classes for instruction in resonance phenomena, scaling, and other methods of analysis including Fourier analysis and Poincaré plots. Using methods from physics on data from human physiology are of particular interest to life science students.

Effective interactions and the hydrogen atom
View Description Hide DescriptionThe technique of effective interactions, which has become commonplace in contemporary particle/nuclear physics, is used to analyze the structure of hydrogen atom energy levels.

Feynman’s proof of Maxwell equations and Yang’s unification of electromagnetic and gravitational Aharonov–Bohm effects
View Description Hide DescriptionFeynman’s unpublished proof of the Maxwell and Lorentzequations, discovered in 1948, was published by Dyson in 1990, together with editorial comments to place the proof into historical context. The proof indicated that the only possible fields that can consistently act on a quantum mechanical particle were gauge fields. Yang’s proof of the Lorentzequation, given in 1983 on the occasion of Tonomura’s experimental confirmation of the Aharonov–Bohm effect, related the electromagnetic and gravitational Aharonov–Bohm effects with the gauge fields. Yang’s discussion, including a remark about Einstein’s objection to Weyl’s original idea of gauge (scale) fields, is given in the context of current research programs for the gauge unification of fundamental forces.

What is a state in quantum mechanics?
View Description Hide DescriptionIn quantum mechanics, states are supposed to be specified by vectors in Hilbert space. However, students become confused about the representation of states and the meaning of “state” itself. We discuss consequences of the fact that quantum mechanics is intrinsically a probabilistic theory, and the ubiquitous confusion over whether quantum states, when specified as well as nature permits, are described by state vectors or rays.

Numerov extension of transparent boundary conditions for the Schrödinger equation in one dimension
View Description Hide DescriptionWe describe an algorithm for animating timedependent quantum wave functions in one dimension with very high accuracy. The algorithm employs the Crank–Nicholson approximation for the time dependence along with a Numerov extension of the discrete transparent boundary conditions described recently by Ehrhardt. We illustrate the power of this approach by simulating the decay of alpha particles from radioactive nuclei and the resonance scattering of electrons in a threelayer GaAs–GaAlAs sandwich.

Creating a distributed physics department
View Description Hide DescriptionTo preserve their undergraduate physics programs, a group of state universities in Texas has jointly offered upper level undergraduate physics courses through the Texas Electronic Coalition for Physics for the past nine years. At the start of the 2002 academic year they formalized their relationship to successfully create a fully functioning distributed physics department. We report on the status of this innovative project, the interinstitutional organizational structures that have evolved to support this initiative, the successes achieved by the distributed department during its first year, and the problems encountered and possible solutions.

An experimental method for studying twodimensional percolation
View Description Hide DescriptionA simple experimental technique for analyzing a broad range of twodimensional percolation problems is presented. The method is based on a combination of the use of a CAD program capable of dealing with a variety of sitebond combinations and an electrical measurement of conductance. The latter is achieved by printing the computer generated pattern using conducting ink. The metalinsulator transition is measured on the print out of the lattice, and the conductivity critical exponent and the percolation threshold are calculated from these measurements.

Unified equations for the slope, intercept, and standard errors of the best straight line
View Description Hide DescriptionIt has long been recognized that the leastsquares estimation method of fitting the best straight line to data points having normally distributed errors yields identical results for the slope and intercept of the line as does the method of maximum likelihood estimation. We show that, contrary to previous understanding, these two methods also give identical results for the standard errors in slope and intercept, provided that the leastsquares estimation expressions are evaluated at the leastsquaresadjusted points rather than at the observed points as has been done traditionally. This unification of standard errors holds when both and observations are subject to correlated errors that vary from point to point. All known correct regression solutions in the literature, including various special cases, can be derived from the original York equations. We present a compact set of equations for the slope, intercept, and newly unified standard errors.

Faraday’s law, Lenz’s law, and conservation of energy
View Description Hide DescriptionWe describe an experiment in which the induced electromotive force in a coil caused by an acceleratingmagnet and the position of the moving magnet are measured as a function of the time. When the circuit is completed by adding an appropriate load resistor, a current that opposes the flux change is generated in the coil. This current causes a magnetic field in the coil which decreases the acceleration of the rising magnet, as is evident from the position versus time data. The circuit provides a direct observation of effects that are a consequence of Lenz’s law. The energy dissipated by the resistance in the circuit is shown to equal the loss in mechanical energy of the system to within experimental error, thus demonstrating conservation of energy. Students in introductory physics courses have performed this experiment successfully.

Jacobi’s 1842 solution of the inversesquare problem
View Description Hide DescriptionIn a memoir written in Latin, Jacobi published in 1842 a simple, concise, and elegant solution of the inversesquare problem. This note argues that Jacobi’s solution has not attracted the attention that it deserves and that it should prove valuable to teachers and students in introductory mechanics courses. A translation of the excerpt in which Jacobi describes his solution is given, and the method is then recast in the now more familiar language of vectors.

Path integral in a magnetic field using the Trotter product formula
View Description Hide DescriptionThe derivation of the Feynman path integral based on the Trotter product formula is extended to the case where the system is in a magnetic field.

The Born–Oppenheimer approximation: A toy version
View Description Hide DescriptionThe Born–Oppenheimer approximation, which is central to the physics and chemistry of molecules and solids, is illustrated by a onedimensional toy model that is easily solved.

Electromagnetic waves in dissipative media revisited
View Description Hide DescriptionWe present a systematic and selfconsistent analysis of the polarization characteristics of monochromatic electromagnetic waves in linear, isotropic, and homogeneous dissipative media. It is shown that in general the transversality property of the waves no longer holds when they propagate in dissipative media. The same is true for electromagnetic waves traveling parallel to the boundary interface if total internal reflection occurs, even in a strictly nondissipative medium. A systematic characterization of elliptically polarized fields for incident s and p waves is presented. Examples in which the time averaged Poynting vector is not parallel to the direction of propagation of the electromagnetic waves are presented. A general expression for the Poynting vector is derived and is interpreted as the power per unit area developed by the electric stress tensor along the direction of propagation of the wave. An equivalent expression is obtained in terms of the magnetic stress tensor with, in general, a different direction of propagation.
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 NOTES AND DISCUSSIONS


The diffraction and spreading of a wavepacket
View Description Hide DescriptionThe spreading of a onedimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the threedimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to nonGaussian wavepackets and to Fraunhoffer diffraction also is discussed.

A new appraisal of old formulations of mechanics
View Description Hide DescriptionLazare Carnot’s formulation of mechanics (1783) is reevaluated in relation to other formulations of classical mechanics and modern theories.
