Volume 76, Issue 9, September 2008
 LETTERS TO THE EDITOR



Landau’s reply
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 PAPERS


Quantum mechanical sum rules for two model systems
View Description Hide DescriptionSum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of onedimensional quantum mechanical sum rules and apply them to the infinite well and the single function potential. These examples illustrate the different ways in which these sum rules can be realized and the varying techniques by which they can be confirmed. We use the same methods to evaluate the secondorder energy shifts arising from the introduction of a constant external field, namely the Stark effect.

Magnetometer construction and applications for introductory physics
View Description Hide DescriptionStudentconstructed magnetometers and commonly available software illustrate the use of autocorrelation functions to extract lowamplitude signals from background noise. The magnetometers are later used to measure forces at sampling rates over , providing data for an introduction to numerical integration.

A tabletop experiment for the direct measurement of the speed of light
View Description Hide DescriptionWe describe a tabletop experiment for the direct measurement of the speed of light to an accuracy of a few percent. The experiment is accessible to students not majoring in science or engineering. The experiment can include a measurement of the index of refraction of a sample. We discuss the experimental apparatus and safety considerations. The results and limitations of the experiment are analyzed, based partly on our experience in employing the experiment in student laboratories.

Summing the molecular contributions to skylight
View Description Hide DescriptionThe skylight that we observe on a clear day is the result of the scattering of sunlight by the gas of the atmosphere. In the literature, two approaches have been used to account for this scattering. The first approach involves the direct summation of the sunlight scattered by the individual molecules of the gas and is the method originally used by Rayleigh. The second approach involves the scattering of sunlight by fluctuations in the relative permittivity of the gas and is the method developed by Smoluchowski and Einstein. In discussions of the origin of skylight, only one of these two approaches is generally followed, and it is sometimes stated or implied that the other approach is inaccurate or not applicable to the scattering of light by the atmosphere. In this paper, a simple model is adopted for the gas in the atmosphere. Then this model is used to obtain the irradiance for the sunlight scattered by the molecules contained in a coherence volume. The irradiance involves the statistical evaluation of a sum, and this sum is taken as the common starting point for investigating the two approaches mentioned before. Simple numerical calculations, based on a random number generator, are used to perform the statistical calculations, and in the end, the observed irradiances are shown to be the same for both approaches.

A simple experiment on slow light in ruby
View Description Hide DescriptionWe describe a simple, reliable, and inexpensive experiment for teaching the basic features of slow light propagation using a solid saturable absorber at room temperature. The experiment is accessible to undergraduate and graduate students. We provide an explanation and analysis based on the coherent population oscillations model, which describes the propagation of light across a ruby crystal.

Forensic Physics 101: Falls from a height
View Description Hide DescriptionThe physics of falling from a height, a topic that could be included in a course on forensicphysics or in an undergraduate class as an example of Newton’s laws, is applied to a common forensic problem.

Complementarity in the Einstein–Bohr photon box
View Description Hide DescriptionThe Bohr–Einstein photon box thought experiment is a forerunner of the EPR experiment: a packet of radiation escapes from a box, and the boxplusradiation state remains entangled. Hence, a measurement on the box makes a difference for the state of the faraway radiation long after its escape. This consequence is analogous to the EPR experiment, but an additional aspect of the Bohr–Einstein box is that complementary magnitudes of the escaped radiation seem correlated to jointly measurable box quantities, which appears to imply a violation of the uncertainty relations. We show how the details of the dynamics make this latter paradox disappear. This calculation makes Bohr’s qualitative arguments in defense of the uncertainty relations precise and reinforces the EPRlike behavior that Einstein later claimed to have had in mind. The calculation reveals that the quantum description of the faraway radiation depends on exactly when the box measurement is performed.

Measuring anharmonicity in a large amplitude pendulum
View Description Hide DescriptionWe investigate the anharmonicity of a large amplitude pendulum and develop a novel technique to detect the high Fourier components as a function of the amplitude for large amplitudes close to 180°. The technique involves doing a Fourier analysis on each halfcycle of the amplitude versus time. The presence of the third and fifth harmonics is detected and the variations of the corresponding Fourier coefficients with amplitude are studied. The experimental setup is inexpensive and simple to implement.

Uniqueness of chemical equilibria in ideal mixtures of ideal gases
View Description Hide DescriptionWe prove the uniqueness of chemical equilibrium for an ideal mixture of ideal gases in a closed, spatially homogeneous volume. Uniqueness, a fundamental issue of chemical physics, is incompletely justified in textbooks and much of the scientific literature. We first reproduce a little known proof by Zel’dovich and show in a more direct fashion than originally presented that a unique equilibrium exists for isothermal reactions. Zel’dovich’s approach is then extended to the adiabatic case, and a more complete exposition than that of Aris is provided. The example of an isothermal, isochoric system provides an illustration of uniqueness. The discussion should be useful for students and instructors of graduate level thermal physics, as well as for researchers in macroscale reactiondynamics.

Novel optical properties of a submerged light bulb
View Description Hide DescriptionWhen an unlit frosted light bulb is immersed in water, the inside frosting appears to shrink, and the outside surface of the bulb appears to become reflective. We explore these interesting effects using geometric optics at the level of introductory physics. It is found that rays emitted from the particles of powder that coat the inside surface of a frosted bulb emerge in conical beams when the bulb is submerged, which results in a reduction of the apparent size of the bulb’s internal surface. The observed reflectivity is explained by the total internal reflection that occurs at the internal glassgas interface. Light rays that refract through a clear bulb are analyzed to understand why a small spherical internal surface appears to exist in this case. The effects of bulb thickness, viewing height, and frustrated total internal reflection are also considered.

The optimal journey from A to B
View Description Hide DescriptionHow fast can you comfortably travel between two points A and B? This question is formulated as a minimization problem of a functional where the discomfort is quantified in terms of the integral of the square of the acceleration between A and B. The problem is solved in terms of the corresponding Euler–Lagrange equation and approximately using a direct variational approach based on trial functions and Ritz optimization. The main purpose of the analysis is to introduce undergraduate students to variational calculus in an interesting and pedagogical way.

The role of correlation functions in the theory of optical wave fields
View Description Hide DescriptionWe discuss the fundamental role of correlation functions in optical wave fields. These functions determine important properties such as the spectrum, the state of polarization, and the state of coherence of light. These properties generally change on propagation, even when the field travels through free space. Two sources which have the same spectrum and the same state of polarization can produce fields whose spectrum and state of polarization are different. These effects can be understood by considering certain correlation functions and the laws which govern their propagation.

Electric fields in steadystate superconductivity according to London theory
View Description Hide DescriptionWe show that a London superconductor in a steady uniform external magnetic field must support an electric field in its interior. The existence of an electric field implies that a superconductor has a nonvanishing charge in its interior, a fact consistent with measurements of charge imbalance in steadystate superconductivity. We model the superconducting state as a collisionless zerotemperature electron gas. The London relation between the supercurrent velocity field and the vector potential is used to show the existence of an induced electric field. We argue that the inhomogeneities in the charge density give rise to a pressure gradient which we describe with the Fermi gas model. By analyzing the electric field, we show that there are two length scales in the system — the London penetration depth and a length associated with the compressibility of the electron gas. We determine the net electric potential in the interior of the superconductor.
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 PHYSICS EDUCATION RESEARCH


A flexible homework method
View Description Hide DescriptionThe traditional methods of assigning and grading homework in large enrollment physics courses have raised concerns among many instructors and students. In this paper we discuss a costeffective approach to managing homework that involves making half of the problem solutions available to students before the homework is due. In addition, students are allowed some control in choosing which problems to solve. This paperbased approach to homework provides more detailed and timely support to students and increases the amount of selfdirection in the homework process. We describe the method and present preliminary results on how students have responded.

Validity and reliability of the force and motion conceptual evaluation
View Description Hide DescriptionThe assessment of learning has become a key component in program evaluation, grant proposals, and education research. Assessment requires valid and reliable instruments. The Force and Motion Conceptual Evaluation (FMCE) is one of several multiplechoice tests used to evaluate the learning of force and motion concepts. Although many physics educationresearchers accept its validity and reliability, validity and reliability estimates based on typical statistical analyses of data have not been established. This study used FMCE posttest results for approximately 150 students in a firstsemester college physics course to estimate reliability and content validity. The results indicate that the FMCE is a valuable instrument for measuring student learning.
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 BOOK REVIEWS


Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin
View Description Hide DescriptionThis article reviews Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin by Lawrence Weinstein, John A. Adam
