Volume 71, Issue 11, November 2003
 PAPERS


Must a Hamiltonian be Hermitian?
View Description Hide DescriptionA consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space–time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a nonHermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PTsymmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.

Direct demonstration of the transverse Stern–Gerlach effect
View Description Hide DescriptionA Stern–Gerlach experiment constructed in the spirit of the original experiment is reported that demonstrates spatial quantization for a beam of silver atoms directly to the unaided eye. Splitting of the atomic beam is seen both parallel to the magnetic field direction, as in the “usual” experiment, and in a direction mutually perpendicular to the field and atomic trajectory, which is a new transverse Stern–Gerlach effect. Splitting of up to 4 mm can be recorded visually. Calculations based on the force components acting on the atomic beams reproduce the experimental results within the experimental accuracy of

Fasterthanlight speeds, tachyons, and the possibility of tachyonic neutrinos
View Description Hide DescriptionFasterthanlight speeds and hypothetical FTL particles known as tachyons are exciting subjects for students, given their speculative and controversial nature. This article presents an overview of these subjects and their role in special relativity and examines the possibility that one or more of the three neutrinos is a tachyon. The paper also describes several low tech demonstrations useful for teaching about fasterthanlight speeds and tachyons in intermediate and advanced introductory collegelevel physics courses.

Nonisochronism in the interrupted pendulum
View Description Hide DescriptionWe experimentally studied the dependence of the period of the interrupted pendulum as a function of the amplitude for small angles of oscillation. We found a new kind of dependence of the period with the amplitude of the pendulum that indicates that if the interruption is not located on the main vertical axis that contains the point of suspension, the period of the interrupted pendulum is highly nonisochronous and does not converge to a definite value as the maximum amplitude approaches zero. We have developed a simple model that satisfactorily explains the experimental data with no adjustable parameters. This property of the interrupted pendulum is a general property of the parabolic potential consisting of two quadratic forms with different curvatures that join at a point different from the apex or the vertex.

Laser interferometric characterization of a vibrating speaker system
View Description Hide DescriptionAn experiment that combines optomechanical and electrical measurements for the characterization of a loudspeaker is presented. We describe a very simple laser vibrometer for evaluating the amplitude of the vibration (displacement) of the speaker cone. The setup is essentially a Michelsontype interferometer operated by an inexpensive semiconductor laser(diode laser). It is shown that the simultaneous measurements of three amplitudes (displacement, electrical current, and applied voltage), as functions of the frequency of vibration, allow us to characterize the speaker system. The experiment is easy to perform, and it demonstrates several useful concepts of optics, mechanics, and electricity, allowing students to gain an intuitive physical insight into the relations between mathematical models and an actual speaker system.

The importance of weightlessness and tides in teaching gravitation
View Description Hide DescriptionWe examine the presentation of the weight, weightlessness, and tides in universitylevel physics textbooks. Introductory textbooks often do not discuss tidal forces even though their understanding would be useful for understanding weightlessness. The explanations of tides often miss the free gravitational motion of both interacting objects, which is essential for the symmetry of tidal deformation. The shortcomings in the explanations of weightlessness and tides as provided by students and teachers are compared to textbook discussions. We suggest that an explicit discussion of the different definitions of weight and a synergetic presentation of weightlessness and tides might lead to a better understanding of gravitation. Our approach is illustrated by examples of tidal effects appropriate for introductory courses.

On perturbation theory in statistical mechanics
View Description Hide DescriptionWe discuss the application of perturbation theory to statistical mechanics both in the operator form of Schwinger and in the scalar form as a sum over states. We briefly consider and compare the use of variational parameters in both approaches. By means of a simple model we show that Schwinger perturbation theory might fail when the eigenfunctions satisfy Dirichlet boundary conditions. On the other hand, the sum over states gives accurate results.

Heat capacity in bits
View Description Hide DescriptionThe temperature may be expressed as the rate of energy increase per unit increase in the state uncertainty under nowork conditions. The consequences of such a choice for heat capacities are explored. I show that the ratio of the total thermal energy to is the multiplicity exponent (log–log derivative of the multiplicity) with respect to energy, as well as the number of base units of mutual information that is lost about the state of the system per fold increase in the thermal energy. Similarly, the nowork heat capacity is the multiplicity exponent for temperature, making independent of the choice of the intensive parameter associated with energy (for example, vs to within a constant, and explaining why its usefulness may go beyond the detection of thermodynamic phase changes and quadratic modes.

How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories
View Description Hide DescriptionImproved models for the pitch, batting, and postimpact flight phases of a baseball are used in an optimal control context to find bat swing parameters that produce maximum range. The improved batted flight model incorporates experimental lift and drag profiles (including the drag crisis). An improved model for bat–ball impact includes the dependence of the coefficient of restitution on the approach relative velocity and the dependence of the incoming pitched ball angle on speed. The undercut distance and bat swing angle are chosen to maximize the range of the batted ball. The sensitivity of the maximum range is calculated for all model parameters including bat and ball speed, bat and ball spin, and wind speed. Postimpact conditions are found to be independent of the ball–bat coefficient of friction. The lift is enhanced by backspin produced by undercutting the ball during batting. An optimally hit curve ball will travel farther than an optimally hit fastball or knuckleball due to increased lift during flight.

On Mach’s critique of Newton and Copernicus
View Description Hide DescriptionBy maintaining the relativity of all motion, especially rotational motion, Mach denied the existence of absolute motion and of absolute space. Accordingly, he maintained the equivalence of the Ptolemaic and the Copernican systems and the equivalence of rotatingsystem/fixeduniverse and universerotating/fixedsystem situations. An analysis of the Foucault pendulum shows that Mach’s relativity principle implies that there cannot be a fixed bucket in a rotating universe. Also, Mach’s views violate the physics that he espoused: noninertial experiments, for example stellar aberration and electromagnetic effects, distinguish between a rotating bucket in a fixed universe and a fixed bucket in a rotating universe, between a Copernican universe and a Brahean or Ptolemaic universe, and establish that one cannot ascribe all pertinent observations solely to relative motion between a system and the universe.

The emergence of semiconductors: Nineteenth century modern physics
View Description Hide DescriptionVolta’s invention of the battery at the end of the eighteenth century generated an era of research devoted to electrical conduction. Classical theory served well for currents in metals, but was contradicted by behavior observed in other materials. Such behavior was labeled anomalous and not understood. Similarities between such substances were known by about 1885; the term Halbleiter (semiconductors) first appeared in the literature in 1911. The twentieth century began with other discoveries that more dramatically contradicted classical theory and drove the development of quantum mechanics. In contrast, the earlier anomalies remained baffling until quantum mechanics could be applied. This converse relationship leads to questions about the coverage of history in modern physics courses.

Computercontrolled inclass feedback system for interactive lectures
View Description Hide DescriptionInstructors who wish to implement an interactive lecture style, such as peer instruction, have the need to collect feedback from students in a lecture environment. We present a computercontrolled electronic circuit that allows for quick, rigorous, and accurate measurement and reporting of student feedback in the lecture environment.

Helping students develop an understanding of Archimedes’ principle. I. Research on student understanding
View Description Hide DescriptionThis paper is the first of two that describe how research on student understanding of Archimedes’ principle is being used to guide the development of instructional materials on this topic. Our results indicate that standard instruction on hydrostatics leaves many science and engineering majors unable to predict and explain the sinking and floating behavior of simple objects. A number of serious and persistent difficulties with the concepts and principles used to analyze such behavior are identified. Although some of these difficulties are specific to the concept of the buoyant force, many others seem to reflect lingering confusion about concepts that are widely assumed to be understood by students before the study of hydrostatics begins.

Helping students develop an understanding of Archimedes’ principle. II. Development of researchbased instructional materials
View Description Hide DescriptionThis paper is the second of two that describe how research on student understanding of Archimedes’ principle is being used to guide the development of instructional materials to supplement instruction on this topic in typical introductory courses. The instructional materials that resulted have proven to be effective. Also discussed are instructional materials for special courses and workshops for K–8 teachers. Evidence is presented that, on some tasks, teachers who have worked through these materials perform much better than introductory physics students.
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 NOTES AND DISCUSSIONS


Huygens’ principle and Young’s experiment in the propagation of light beams
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Average distance between a star and planet in an eccentric orbit
View Description Hide DescriptionThe average distance between a star and a planet in the classic twobody problem of celestial mechanics is different from the orbital semimajor axis of the planet if the average is performed over time or angle. The time and angleaveraged distances are functions of both the semimajor axis and the orbital eccentricity and can differ from the orbital semimajor axis by as much as 50%–100%. This difference contradicts the usual statement found in most introductory and advanced textbooks that the average distance and the orbital semimajor axis are equal.

Torque and the rate of change of angular momentum at an arbitrary point
View Description Hide DescriptionIn this article we show that the general relation between torque and rate of change of the angular momentum for an arbitrary point lets us obtain explicit solutions of many problems in plane motion of a rigid body when the motion of one of its points is given. This method involves considerably less calculation than the standard procedure.
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 APPARATUS AND DEMONSTRATION NOTES


Mechanics of the slow draining of a large tank under gravity
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A basic lockin amplifier experiment for the undergraduate laboratory
View Description Hide DescriptionWe describe a basic experiment for the undergraduate laboratory that demonstrates aspects of both the science and the art of precision electronic measurements. The essence of the experiment is to measure the resistance of a small length of brass wire to high accuracy using a simple voltage divider and a lockin amplifier. By performing the measurement at different frequencies and different drive currents, one observes various random noise sources and systematic measurement effects.
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 BOOK REVIEWS


Biophysics: An Introduction
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