bluuebunny/crossref_metadata_2025 huggingface dataset:
Nonreduction of the state vector in quantum measurement
Authors: Fox J. R.
Published: March 2005

We describe a formalism which allows us to regard the quantum-mechanical state vector as developing solely according to the Schrödinger equation and not being subject to any reduction or collapse upon measurement. On the other hand, our relationship, as observers, to the state is revised upon our learning the results of measurements. By introducing joint probability operators, whose expectation values in the unreduced state yield the probabilities of obtaining specified measurement results sequentially, we are able to reproduce the predictions of orthodox quantum theory for all testable quantities.
A simple experiment for determination of magnetic anisotropy
Authors: Ribeiro P. Costa , Bemski G. , Davidovich M. A. , Wajnberg E.
Published: March 2005

While experiments involving measurements of magnetic susceptibility in undergraduate laboratories are quite common, it is perhaps not fully appreciated that the anisotropy of magnetic susceptibility can be measured with the use of a small magnet, a microscope, and a stopwatch. This problem is discussed in relation to an illustrative example which involves the orientation of sickle cells in the magnetic field.
Effect of blackbody radiation on the hydrogen spectrum
Authors: Yorke Ellen D.
Published: March 2005

The partition function of a single hydrogen atom diverges. While the astrophysical solution to this dilemma is the introduction of a density dependent alteration of the partition function, the question of what happens in an arbitrarily rarefied gas remains open. It is argued that in this case blackbody radiation changes the atom’s spectrum. A unified derivation including these effects and the nonrelativistic Lamb shift and spontaneous emission is presented. Then recent numerical calculations performed by workers in the field of Rydberg atom spectroscopy are utilized to estimate the principal quantum number for which blackbody radiation significantly alters the spectrum.
Petrov classification: An elementary approach
Authors: Kalotas T. M. , Eliezer C. J.
Published: March 2005

We reduce the length of the usual algebraic classification scheme of the Weyl tensor by avoiding the step centered around its reduction to canonical form. Instead the different algebraic types are established more economically via the elementary approach of constructing explicit examples.
Relativistic one-dimensional hydrogen atom
Authors: Lapidus I. Richard
Published: March 2005

The energy eigenvalues and eigenfunctions of the relativistic one-dimensional hydrogen atom with a delta function interaction are obtained by solving the Dirac equation. The single bound state is compared to the ground state of the relativistic three-dimensional hydrogen atom.
Determining the age of gas lantern mantles using gamma-ray analysis
Authors: Luetzelschwab John
Published: March 2005

The thorium used in gas lantern mantles is a naturally radioactive element. Using the buildup and decay of the daughters in the thorium decay series, the age of a gas lantern mantle can be determined.
Elementary approach to scattering theory
Authors: Haller Kurt
Published: March 2005

Expressions for the time-dependent scattering wave function, S-matrix elements, and the transition rate are derived using simple time-dependent perturbation theory as it is presented in elementary quantum-mechanics texts.
A recursive method for solving the Schrödinger equation
Authors: Shakir Sami A.
Published: March 2005

A recursion equation for computing the total reflectance amplitude from a given potential is derived. It is shown that the poles of the total reflectance amplitude are the bound eigenvalues of the problems. The method avoids complicated boundary value problems usually encountered in computing the bound eigenvalues of the Schrödinger equation. The utility of the method is demonstrated by two examples. A simple algorithm for the numerical approach is derived. Finally the physical meaning in terms of standing waves is discussed.
Inexpensive demonstration of an anharmonic oscillator
Authors: Smith H. J. T. , Woolner K. A.
Published: March 2005

A simple circuit comprising an operational amplifier and an RC feedback network is configured into a resonant circuit. It behaves as a simple harmonic oscillator for small input signals and changes to an anharmonic oscillator when slew limiting occurs. The circuit is analyzed and the theoretical gain is compared to experimental data for the two cases.
Experiments on diffusion in liquids using holographic interferometry
Authors: Fenichel Henry , Frankena Hans , Groen Fokke
Published: March 2005

An experiment is described which uses the technique of holographic interferometry to study diffusion in liquids. The diffusion process can be recorded on double exposed holograms or it can be observed and recorded in real time using video techniques.
For whom the bell tolls
Authors: Kunstatter G. , Trainor L. E. H.
Published: March 2005

We attempt to clarify the relationship between the Einstein–Podolsky–Rosen analysis and Bell’s theorem in the context of the statistical interpretation of quantum mechanics. We conclude that despite certain claims, the statistical interpretation cannot reconcile the EPR notion of a local, objective reality with the prediction of quantum mechanics.
Computers in physics education: An example dealing with collision phenomena
Authors: Borghi L. , De Ambrosis A. , Mascheretti P. , Massara C. I. , Gazzaniga G. , Ironi L.
Published: March 2005

This paper describes an interactive graphics package for teaching and learning collision phenomena which includes analysis of one-dimensional elastic collisions, two-dimensional elastic collisions, and inelastic collisions. The package is intended as a support for a traditional course for first-year physics students.
The receiving antenna
Authors: Benumof Reuben
Published: March 2005

The basic theory of a receiving antenna is presented in sufficient detail to permit the calculation of the voltage across the load of an optimally designed TV antenna circuit. The rms load voltage is given by VL=λH(30RrD)1/2, where λ is the wavelength of the incident radiation, H is the rms value of the magnetic field intensity, Rr is the radiation resistance of the antenna, and D is the directive gain. The result obtained from this formula for a typical monopole TV antenna is compared with that obtained from a simple, intuitive approximation. The reasons for the difference in the results are discussed.
A selective-surface solar radiometer
Authors: Fendley Jack , Hutchins Michael
Published: March 2005

A solar radiometer of novel design was used to measure the horizontal global solar irradiance. The instrument employs the principle of spectral selectivity of surfaces and gives directly an absolute determination of the solar irradiance. Construction is simple and the device is suitable for use in colleges.
Andrei Sakharov: A man of our times
Authors: Sessler Andrew M. , Howell Yvonne
Published: March 2005

The following is based upon a talk given at the American Physical Society Meeting in Baltimore, 18–21 April 1983 entitled, ‘‘Sakharov and Society’’ by Andrew M. Sessler. The occasion was the presentation by the Forum on Physics and Society of the 1982 Leo Szilard Award to Andrei Sakharov who was, of course, unable to attend. Dr. Sessler, former Director of the Lawrence Berkeley Laboratory, was Chairman, during 1982, of the American Physical Society Committee on the International Freedom of Scientists. Yvonne Howell is a freelance writer.
Helmholtz theorem for antisymmetric second-rank tensor fields and electromagnetism with magnetic monopoles
Authors: Kobe Donald H.
Published: March 2005

A generalized Helmholtz’s theorem is proved, which states that an antisymmetric second-rank tensor field in 3+1 dimensional space-time, which vanishes at spatial infinity, is determined by its divergence and the divergence of its dual. When the divergence of the antisymmetric electromagnetic field strength tensor is equal to the electric charge-current density and the divergence of the dual of the electromagnetic field strength tensor is equal to the magnetic charge-current density, the equations of electromagnetism are obtained. As a convenience, in the solution of the equations of electromagnetism two different four-vector potentials can be used, one of which couples to the electric charge-current density and the other to the magnetic charge-current density.
On the recurrence phenomenon of a resonant spring pendulum
Authors: Lai H. M.
Published: March 2005

Olsson’s coupled equations for a resonant spring pendulum are solved. To-and-fro transfer of energy from one mode to another is shown, the period for such a recurrence is derived, and the patterns of the mass trajectory are obtained. Experimental data are reported and good agreement is found with the theory.
A note on quantum and cosmological mass relations
Authors: Sharma M. L. , Sharma N. K.
Published: March 2005

Limits on elementary particle masses are obtained from relations among the fundamental constants G, ℏ, c, and H0. (AIP)
Resonance formation and decay: An application of second-order, time-dependent perturbation theory
Authors: Norbury J. W. , Deutchman P. A.
Published: March 2005

Using second-order, time-dependent perturbation theory, we calculate the intermediate and final state amplitudes for transitions from an initial to a final state via an intermediate resonance. When our complex energies are replaced by real energies our results are identical to the real energy final state amplitudes obtained by other authors, thus justifying the usual phenomenological replacement of real energies with complex ones when considering resonance behavior. We then derive the Fermi Golden Rule No. 1 with complex energies.
Magnetic pendulum apparatus for analog demonstration of first-order and second-order phase transitions and tricritical points
Authors: Schmidt V. Hugo , Childers Bryan R.
Published: March 2005

A simple apparatus is described in which the deflection of a pendulum with an electromagnet mounted at its tip is the analog of the magnitude of the order parameter in a phase transition. Its deflection is governed by the current through this electromagnet and currents through fixed electromagnets just below it. By suitable variation of these currents, the pendulum can be made to undergo gradual (second-order transition analog) or sudden (first-order transition analog) displacements from the vertical, as well as the intermediate situation corresponding to a tricritical point. Results for critical and tricritical exponents obtained with this apparatus are presented.
Suppression and restoration of constants in physical equations
Authors: Desloge Edward A.
Published: March 2005

Theoretical calculations are frequently simplified by setting some of the constants which occur equal to unity. The theory behind this practice is considered. It is shown that if we start with a set of dimensionally homogeneous equations, it is possible, by reducing the set of equations to an equivalent set of dimensionless equations, to assign arbitrary values to any set of dimensionally independent constants occurring in the equations. It is then shown how, after arbitrary mathematical manipulations on the resulting simpler set of equations, the equations can at any stage be restored to dimensional form with the resultant recovery of the suppressed constants.
Aerodynamic drag crisis and its possible effect on the flight of baseballs
Authors: Frohlich Cliff
Published: March 2005

At Reynolds numbers above about 105 the aerodynamic drag force on a sphere drops sharply as the flow begins to become turbulent in the boundary layer. For baseballs, this ‘‘drag crisis’’ may occur at speeds which are typical for pitched or batted balls. The effects of the drag reduction on the behavior of both pitched and batted balls is significant, and may explain several features of the game of baseball which previously have been unexplained or attributed to other causes. In particular, the drag reduction may help to explain why pitched fastballs appear to rise, why pitched curve balls appear to drop sharply, and why home run production has increased since the introduction of the alleged ‘‘lively ball.’’ Calculations suggest that aerodynamic forces are as important a factor in fastpitch softball as in baseball, and that they are a critical factor in a number of other ball games.
Status of the Copernican theory before Kepler, Galileo, and Newton
Authors: Martin Douglas R.
Published: March 2005

The most obvious evidence supporting the Copernican theory emerged during a period of 300 years following the publication of De Revolutionibus by Copernicus in 1543. In this paper a comparison is made between the geo-centric and helio-centric theories on the basis of astronomical merit and observational evidence available in 1543. The Copernican theory is not superior on the basis of calculational accuracy, but does possess greater unity and is simpler in the sense that ad-hoc assumptions required by the geo-centric theory are not required by the Copernican theory. The Copernican theory, however, requires an ad-hoc assertion about the size of the universe in order to explain the absence of observed stellar parallax.
Relativity without light
Authors: Mermin N. David
Published: March 2005

The relativistic addition law for parallel velocities is derived directly from the principle of relativity and a few simple assumptions of smoothness and symmetry, without making use of the principle of the constancy of the velocity of light.
Solar temperature at 4 GHz: An undergraduate experiment
Authors: Lo George , Lonc William P.
Published: March 2005

An experiment is described using readily available satellite television receiving equipment to obtain an approximate value for the solar (quiet Sun) temperature at a frequency of 4 GHz (7.5 cm), using a comparison method. The procedure described here appears to be quite suitable for use at the undergraduate level inasmuch as it does not require any special electronic equipment. It is found that the observed value is within 10% of the accepted value (approximately 28 000 K). The results are considered to be satisfactory, given the substantial approximations made in the performance of the experiment and analysis of the data.
On the numbers of things and the distribution of first digits
Authors: Lemons Don S.
Published: March 2005

I answer the question ‘‘Why are there more small things in the world than large things?’’ in terms of a probabilisitc model of partitioning a conserved quantity. Benford’s empirical rule for lists of numbers, that the proportion of numbers with first digit m is log10(m+1)−log10 m, is an exact consequence of the model.
Science and society test IX: Technical means of verification
Authors: Hafemeister David
Published: March 2005

Technical means of monitoring are used to verify compliance to arms control treaties. Numerical estimates are developed in order to quantify some aspects of (1) nuclear testing (signatures, optical double pulse calibration, decoupling in cavities, and high-frequency components in seismic signals), (2) optical reconnaissance (films versus CCDs, air turbulence, adaptive optics, and digital image processing), (3) infrared (resolution and detection sensitivity), and (4) radar (synthetic aperture and missile coefficient).
No rumors of transcendence in physics
Authors: Rosen Joe
Published: March 2005

Transcendent and nontranscendent world conceptions are compared. It is shown that five points in physics, claimed to be indicative of a transcendent domain, are consistent with nontranscendence.
Fluid dynamic and kinetic theory models for a nonprovocative land defense of central Europe
Authors: Saperstein Alvin M.
Published: March 2005

A major problem in using theory to make predictions in a practical situation is the validity of the theory. The theoretical predictions are only as good as the theory. A model is developed, based upon a one-dimensional equation of continuity, for predicting the effectiveness of an attritional mode of defense against a conventional land attack in Europe. The validity of the model and its predictions is tested by comparing the results with those of a stochastic, discrete model of multiple small defensive battles, with the latter results obtained by Monte Carlo type computer simulations. The concordance of the results supports the validity of the models and the efficacy of applying physical-type reasoning to pressing, nonphysical, world problems.
Resource letter PS-1: Physics of sports
Authors: Frohlich Cliff
Published: March 2005

This Resource Letter provides a guide to the literature on the physics of sports. The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of a somewhat more specialized nature; and the letter A indicates rather specialized or advanced material. An asterisk (*) indicates those articles to be included in an accompanying Reprint Book.
Acceleration of a buoyant sphere in dry water
Authors: Crawford Frank S.
Published: March 2005

It is pointed out that the formula for the upward acceleration of a submerged, buoyant sphere should contain the effective mass of the water that is accelerated downward. (AIP)
Helmholtz’s theorem revisited
Authors: Kobe Donald H.
Published: March 2005

Helmholtz’s theorem for three-vectors is applicable to electrostatics and magnetostatics, but it must be generalized to antisymmetric second-rank tensors to be applicable to electromagnetism. A proof of Helmholtz’s theorem for three-vectors, which is shorter than the usual one, is given. Helmholtz’s theorem is shown to be a special case of the Hodge decomposition theorem in the theory of differential forms.
On an apparent paradox in the motion of a smoothly constrained rod
Authors: Namias Victor
Published: March 2005

The motion in a vertical plane of a straight rod constrained to move between two frictionless concentric circles is investigated. An apparent paradox arises because the reaction forces at the contact points are purely radial, and together with the rod’s weight, which acts at the center of mass, are uncapable of supplying the torque about the center of mass necessary for the rod’s rotation. A tentative explanation has been given by Carroll [Am. J. Phys. 52, 1010 (1984)], where a mysterious tangential contact force arising from an infinite radial contact force is invoked. We disagree with these conclusions. In postulating that the motion takes place according to certain prescribed geometrical constraints, one must always make certain that the laws of physics are respected, and the model must be chosen accordingly. A simple model consisting of an H-shaped frame with four contact points is used to determine the nature of the contact forces and the origin of the torque. As the link between the two half-rods of the H frame is reduced, a single rod is obtained which can be thought of as the limit of a sector whose angular width is made vanishingly small. The dilemma is resolved by showing that a singular angular density of contact force distribution prevails, but that the contact force itself is finite.
Galileo’s physical measurements
Authors: Drake Stillman
Published: March 2005

Galileo derived the pendulum law and law of fall from careful timings recorded in his working papers of 1604. His units made g′=π2/8 within 1/2 %, the ratio of distance fallen to length of pendulum timing fall by swing to the vertical. This paper extends my ‘‘Galileo and mathematical physics’’ in Scienza e Filosufia, edited by C. Mangione (Garzanti, Milan, 1985).
Proof of the quantum virial theorem
Authors: Dongpei Zhu
Published: March 2005

Based on the Feynman–Hellmann theorem, a new proof of the quantum virial theorem is presented. The same approach is also used to derive the virial formulation in relativistic quantum mechanics and for the case where the vector potential exists.
Simple percolation experiment in two dimensions
Authors: Mehr Ramit , Grossman Tal , Kristianpoller N. , Gefen Yuval
Published: March 2005

A simple experiment for the demonstration of percolation problems is reported. Following the method of Last and Thouless [B. J. Last and D. J. Thouless, Phys. Rev. Lett. 25, 1719 (1971)], measurements were performed on a sheet of conducting paper after randomly cutting out small square ‘‘sites’’ at a concentration 1−p. The conductance and the area fraction covered by the ‘‘infinite cluster’’ P were then evaluated as a function of p. This experiment, carried out at an undergraduate physics laboratory, appears to be appropriate as an introduction to the teaching of phase transitions.
At what altitude does the horizon cease to be visible?
Authors: Bohren Craig F. , Fraser Alistair B.
Published: March 2005

At altitudes greater than a few kilometers the horizon is indistinct because of contrast reduction, even on extraordinarily clear days. An airplane passenger flying over an ocean cannot point to the apparent boundary between earth and sky and confidently proclaim its distance. To determine the distance to the horizon to within 10% requires knowing its angular position to within a few minutes of arc. This is unattainable in a realistic atmosphere, even one that is very clean. Scattering by the air molecules themselves is not quite sufficient to blur the horizon, but it does not take many particles before the horizon becomes indistinct.
Bell’s theorem and the features of physical properties
Authors: Dotson Allen C.
Published: March 2005

Several discussions of the testing of Bell’s theorem have emphasized one possible implication of the test results, namely, that nature at the atomic level is nonseparable. This paper seeks to clarify an alternative approach, which is inherent to the Copenhagen interpretation of quantum mechanics. In particular, it emphasizes the distinction between what we call ‘‘physical properties’’ and what could justifiably be termed ‘‘objective attributes of physical reality.’’
Vector products and pseudovectors
Authors: Hauser Walter
Published: March 2005

A simple way of introducing the coordinate dependence of the vector product of two vectors in introductory courses is presented. Its use for the classification of vector quantities as polar or axial vector quantities is discussed.
On the derivation of ℏ ⋅ d k/d
t=F, the k-space form of Newton’s law for Bloch waves
Authors: Kroemer Herbert
Published: March 2005

A new derivation of ℏ ⋅ dk/dt=F is given that is both rigorous and comparatively simple. The central idea is to consider the time evolution of the expectation value of the translation operator of a state that is initially a pure Bloch wave, rather than attempting to solve the more complicated but unnecessary problem of the actual evolution of the entire wave function. Because interband transitions take place between states that have the same expectation value of the translation operator, a consideration of the time evolution of the latter makes it unnecessary to consider interband transitions at all.
Current transformers
Authors: Cross Rodney C.
Published: March 2005

Current transformers are used primarily to measure time varying currents under conditions where some degree of isolation from the primary circuit is desired. In this paper, the basic physics of current transformers is considered and several useful techniques for extending their performance are described. They can be used in an undergraduate laboratory as the basis for very interesting and illustrative experiments on measurement techniques and on the properties of magnetic materials.
Effects of circular birefringence on light propagation and reflection
Authors: Silverman M. P. , Sohn R. B.
Published: March 2005

A gyrotropic (i.e., circularly birefringent) medium gives rise to a variety of interesting optical effects in addition to the well-known one of optical activity, the rotation of a plane of linear polarization. This article considers manifestations of gyrotropy associated principally with light reflection from a transparent chiral medium. Such effects include the possible inequality of the angles of reflection and incidence, the generation of elliptically polarized light from incident linearly polarized light, and the differential reflection of right and left circularly polarized light. Reflection from a transparent gyrotropic medium is similar in some ways to, although distinct from, reflection from a conductor or absorbing medium.
A simple conceptual model for two-photon absorption
Authors: Moscatelli Frank A.
Published: March 2005

The process of two-photon excitation of atoms is explained by a simple conceptual model which takes into account virtual states of the atom as well as photon statistics. In addition to predicting the correct dependence of the two-photon excitation probability on the intensity of incident light, the model also provides an explanation for a number of experimentally observed effects.
Chaotic dynamics of a bouncing ball
Authors: Tufillaro N. B. , Albano A. M.
Published: March 2005

An undergraduate experiment is described that illustrates the period doubling route to chaos in a simple dissipative mechanical system, a bouncing ball subject to repeated impacts with a vibrating table.
A retrograde motion demonstration model
Authors: Ackroyd Albert , Gerritsen Hendrik J.
Published: March 2005

The construction of an inexpensive, simple machine that demonstrates retrograde motion of inferior and superior planets is described. The motion is made visible by connecting two independently clock-motor driven balls, representing two planets, with a rod indicating the direction vector. The rod has a red and green LED on each end, respectively, to represent retrograde motion of either an inferior or superior planet. Some suggestions for student exercises are included.
Predicting charmonium and bottomonium spectra with a quark harmonic oscillator
Authors: Norbury J. W. , Badavi F. F. , Townsend L. W.
Published: March 2005

We present a simple application of the three-dimensional harmonic oscillator which should provide a very nice particle physics example to be presented in introductory undergraduate quantum mechanics course. The idea is to use the nonrelativistic quark model to calculate the spin-averaged mass levels of the charmonium and bottomonium spectra.
Pendulum damping
Authors: Squire Patrick T.
Published: March 2005

A mathematical model of pendulum damping is presented. The model includes three damping terms, quadratic and linear in the velocity, and velocity independent. The model is applied to detailed experimental data obtained with a computer-linked rigid pendulum. Values of the damping parameters are obtained using nonlinear least squares. It is shown that linear models are inadequate to account for the observed behavior of real pendulums, and that at least two damping terms are generally needed.
Measurement of the transition temperature of a high T
c superconductor
Authors: Fox John N. , Rustad Frederick A. , Smith Roger W.
Published: March 2005

The transition temperature of a ceramic superconductor, Y1Ba2Cu3O7−x, is determined using a tunnel diode oscillator.
The Compton effect: As the ‘‘progenitor’’ of quantal concepts (including the uncertainty principles)
Authors: Akyüz R.Ömür
Published: March 2005

It is shown that, taken as an empirical fact, the Compton effect may directly lead to the Planck–Einstein and Heisenberg uncertainty relations, by an elementary corpuscular assumption and essentially obvious stipulations on the natures of electrons and monochromatic electromagnetic radiation.
Using Einstein’s method to derive both the Planck and Fermi–Dirac distributions
Authors: Crawford Frank S.
Published: March 2005

Many quantum mechanics textbooks give the remarkable argument of Einstein wherein he introduces his famous ‘‘A and B coefficients’’ and uses detailed balancing to derive the Planck distribution for thermal photons. But it seems to be not widely known that the same approach can yield the Fermi–Dirac distribution for thermal electrons.
Spatial dependence of thermoelectric voltages and reversible heats
Authors: Walstrom P. L.
Published: March 2005

Thermocouples are probably the most widely used temperature sensors in science and industry but perhaps also the most often misunderstood. The basic distinction between an ordinary resistance thermometer, in which the sensing element is localized, and a thermocouple circuit, in which potentials vary along the wires themselves, is most clearly shown by a treatment in which the spatial dependences of the phenomena are explicitly invoked. Such treatment is essential to analysis of measurement errors in thermocouple circuits. In this article, the basic equations of thermoelectricity are reviewed and their physical meanings discussed. The profile of the electrochemical potential for an illustrative example is then calculated from published data. Finally, the problem of a thermocouple in a static magnetic field is discussed.
Discontinuity of the electromagnetic fields, potentials, and currents at fixed and moving boundaries
Authors: Namias Victor
Published: March 2005

Expressions for the discontinuity of the electromagnetic field at a fixed boundary such as the interface between two different media are usually derived through the use of Maxwell’s equations in integral form, i.e., Gauss’ law, Faraday’s law, and Ampère’s law. When the boundary is moving, the derivation of the corresponding jump condition becomes considerably more difficult, requiring the use of generalization of Maxwell’s equations in integral form for moving media. Obtaining the discontinuity for the electromagnetic currents is even more intricate and necessitates the use of a kinematic theorem related to the Helmholtz vector flux theorem. This article presents a new technique that yields all the results directly from Maxwell’s equations in their differential form. The method is very straightforward and essentially the same for both fixed and moving boundaries. Elementary operations of vector analysis are all that is basically required to obtain the discontinuity of any electromagnetic quantity, be it a field, a potential, or a source.
Fermion assemblies: From collapse to saturation
Authors: Pacheco Amalio F. , Sañudo Jaime
Published: March 2005

Through a very simple analysis, the condition of stability of a fermion assembly at zero temperature will be studied. The radius of equilibrium is calculated by minimizing the energy of a uniform-density spherical distribution. Depending on the nature of the attractive force existing between the particles, the article will illustrate the different types of collapse induced in the assembly. The property of saturation when a hard core is included will also be shown. These considerations provide simple models for the atom, for the nucleus, and for gravitational clusters of fermions.
Boundary-induced perturbations
Authors: Morris J. R.
Published: March 2005

A simple quantum mechanical system consisting of a particle in a one-dimensional impenetrable box is considered in a situation wherein the wavefunction is subjected to moving boundary conditions. A method of approximating the wavefunction is presented and applied to a simple example.
Power series approximation to solutions of nonlinear systems of differential equations
Authors: Fairén Victor , López Vicente , Conde Luis
Published: March 2005

A useful method for constructing a power series solution to a system of nonlinear ordinary differential equations is explained, based on the transformation of the system to polynomial format. Upon substitution in the polynomial system, the coefficients of the power series are obtained through easily programmable recurrence relations.
The bed of nails revisited
Authors: Bucher Manfred
Published: March 2005

The popular bed of nails demonstration is analyzed by making use of a novel collision diagram. Cases of hitting rubber bricks or cinder blocks illustrate the difference between elastic and inelastic collisions. While the analysis suggests several safety measures, a considerable degree of danger remains.
The force on a magnetic dipole
Authors: Boyer Timothy H.
Published: March 2005

The classical magnetic force on a magnetic dipole depends upon the model for the dipole. The usual electric current loop model for a magnetic dipole leads to the force F=∇(m⋅B) on a magnetic dipole m in a magnetic field B. The separated magnetic charge model for a magnetic dipole leads to the force F=(m⋅∇)B on a magnetic dipole. The latter expression is analogous to the force experienced by an electric dipole in an electric field. Here, some elementary examples are given where the force expressions yield entirely different forces on a magnetic dipole. Electromagnetism textbooks usually do not emphasize the difference between these force expressions; however, occasionally the difference is important for understanding experimental results. In the 1930s and 1940s the difference in force expressions was involved in a determination of the nature of the neutron dipole moment. At present, in the 1980s, the difference in the force expressions is central to a controversy over an experiment to test the proposed Aharonov–Casher effect.
The refractivity of helium and its measurement by laser interferometry
Authors: Hey J. D. , Driver H. S. T. , Fish D. B.
Published: March 2005

This article covers the historical background to the study of the helium atom by optical methods, and some essential results from the classical and quantum mechanical theory of optical dispersion. A simple undergraduate-level experiment, designed for refractivity measurements by cw laser interferometry, is described, and the ingredients of the theoretical calculation of the refractivity are discussed. It is shown that careful experimental work in this area can yield information that provides a valuable test of the accuracy and completeness of the available oscillator strength data.
Numerical experiments in quantum physics: Finite-element method
Authors: Searles Debra J. , von Nagy-Felsobuki Ellak I.
Published: March 2005

A new numerical approach (finite-element method) to solving one-dimensional Schrödinger problems is discussed. Its effectiveness is demonstrated in solving the nontrivial finite well and Morse potential problems. Moreover, it is shown by a case study of 16 low-lying vibrational states of Li2 that the finite-element method can achieve the same order of accuracy more easily than the better known Numerov–Cooley method.
A simple way to understand the origin of the electron band structure
Authors: Cota Ernesto , Flores Jorge , Monsivais Guillermo
Published: March 2005

How the electron band structure appears in a one-dimensional system using simple and intuitive arguments is discussed. The article presents both numerical and qualitative results.
Henry Cavendish, Johann von Soldner, and the deflection of light
Authors: Will Clifford M.
Published: March 2005

The gravitational deflection of light based on Newtonian theory and the corpuscular model of light was calculated, but never published, around 1784 by Henry Cavendish, almost 20 years earlier than the first published calculation by Johann Georg von Soldner. The two results are slightly different because, while Cavendish treated a light ray emitted from infinity, von Soldner treated a light ray emitted from the surface of the gravitating body. At the first order of approximation, they agree with each other; both are one-half the value predicted by general relativity and confirmed by experiment.
Semiclassical treatment of the double well
Authors: Holstein Barry R.
Published: March 2005

The double well potential V(x)= 1/4  λ(x2−α2)2 is addressed using both semiclassical path integral and instanton techniques. The basic physics of the two-state system is shown to arise, and energy levels calculated via the two methods are compared.
Comments on the one-dimensional hydrogen atom [Am. J. Phys. 2
7, 649 (1959); Am. J. Phys. 3
7, 1145 (1969); Am. J. Phys. 4
4, 1064 (1976)
Authors: Hammer C. L. , Weber T. A.
Published: March 2005

Extraneous solutions to the Schroedinger equation are discussed. (AIP)
The aerodynamics of tennis balls—The topspin lob
Authors: Štěpánek Antonín
Published: March 2005

A general description is presented of the calculation of the ballistic trajectory of a flying spinning ball acted on, in addition to the forces of gravity and drag, by the so-called Magnus force. By applying the regression analysis to results of wind-tunnel measurement of the drag and lift coefficients of a spinning ball, a calculation of the nonlinear differential equation of the hodograph was carried out by means of the Runge–Kutta method. The theoretical results that can be used to calculate the ballistic trajectories for any ball game were applied to one of the most difficult and most interesting tennis strokes, i.e., to the topspin lob. Practical results obtained for various distances are presented in a table as well as in graphical form. UFAJP
Teaching special relativity through a computer conference
Authors: Smith Richard C.
Published: March 2005

A recent seminar in special relativity is described, which was taught exclusively through a computer conference, hosted on a distant mainframe computer, and asynchronously accessed by students and instructor with microcomputer and modem. Nine participants offered more than 400 separate discussion contributions over the 13-week span of the course. Criteria for choosing courses to be offered in this mode are suggested, and problem areas that need attention in the conduct of subsequent courses are pointed out.
Did Einstein really discover ‘‘E=m
c2’’?
Authors: Fadner W. L.
Published: March 2005

In 1952, Herbert Ives claimed that Einstein’s first development of E=mc2 was circular, and that he had not been the first to develop that equation. That allegation has been repeated in several more recent works. Earlier, Planck asserted that one of the postulates that Einstein had used in that development was not exact. Those claims and subsequent papers concerning them are examined herein. The surprisingly long history of the mass–energy relation is summarized. In the context of this topic, it is argued that circularity is seldom a legitimate critique of scientific proposals. A simple refutation of Planck’s claim is also included.
A calculus of variations demonstration: The gradient index lens
Authors: Jones Kevin M. , Lundgren Scott , Chakravorty Alak
Published: March 2005

Calculating ray paths in a gradient index lens provides a simple, practical application of the calculus of variations. Inexpensive lenses are available for use as a lecture demonstration.
Comment on ‘‘Periastron precession in general relativity’’ [Am. J. Phys. 5
5, 85 (1987)]
Authors: Ganea I. M.
Published: March 2005

It is shown that the derivation of periastron precession can be improved if the exact general relativistic definition of a circular orbit is used. (AIP)
Gamow vectors and decaying states
Authors: Bohm A. , Gadella M. , Mainland G. Bruce
Published: March 2005

Gamow vectors are generalized eigenvectors of the Hamiltonian with complex eigenvalues that describe exponentially decaying (or growing) states. The energy wavefunctions corresponding to Gamow vectors have a pole immediately below (or above) the real axis in the complex energy plane. Although complex energy values were introduced more than half a century ago for the theory of alpha decay, they have become disreputable and have been banished from quantum mechanics textbooks because of mathematical problems and incorrect physical interpretation. Developments in modern mathematics have now provided a mathematical foundation that has led to the correct physical interpretation. In this article, energy eigenvectors with complex eigenvalues are first introduced by explicitly considering a specific simple decaying system. Then, an elementary account of the properties of Gamow vectors is given. The results from the decaying system provide a motivation for the introduction of Gamow vectors and some of the background information needed to understand their properties.
Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field
Authors: Desloge Edward A.
Published: March 2005

A general expression is obtained for the space-time interval between neighboring events in a one-dimensional space in which it is possible to set up a rigid reference frame. Particular expressions are then obtained for the interval for the special cases of a rigid frame at rest in a uniform gravitational field and a rigid frame uniformly accelerating in field-free space. The two expressions are not equivalent and are used to show why, how, and to what extent observations made in a rigid enclosure at rest in a gravitational field are not equivalent to observations made in a rigid enclosure that is uniformly accelerating in field-free space. Two facts of particular interest that are demonstrated in the course of the analysis are the following: (i) Two spatially separated particles that are simultaneously released from rest and allowed to fall freely in a uniform gravitational field will not remain at rest with respect to one another. (ii) Uniformly accelerating reference frames and inertial frames are the only possible one-dimensional rigid frames in flat space-time.
On the principles of quantum mechanics and the reduction of the wave packet
Authors: Balian Roger
Published: March 2005

The principles of quantum mechanics are reviewed in a form where the concept of state appears both more transparent and more general than in current textbooks. The so-called reduction of the wave packet is then derived from some of these principles (such as the compatibility of commuting observables and the fact that a variance cannot be negative), instead of being posed as a separate postulate.
Harriet Brooks—Pioneer nuclear scientist
Authors: Rayner-Canham M. F. , Rayner-Canham G. W.
Published: March 2005

This article, using revealing statements from contemporary correspondence, traces the eventful life of Harriet Brooks, one of Ernest Rutherford’s most valued research students and collaborators at McGill University. Brooks performed some of the crucial experiments in the early work on radioactivity; her work led her to the Cavendish where she did work with J. J. Thomson. Still later, she worked with Marie Curie, to whom Rutherford favorably compared her. Despite Brooks’ achievements and promise, she finally relinquished her research career when faced with insurmountable objections to women who wished to have both a professional and a married life.
Unified diagram for Fraunhofer diffraction and interference patterns from single, double, and multiple slits
Authors: Bucher Manfred , Siemens Duane P.
Published: March 2005

A unified diagram is presented, based on the Huygens–Fresnel principle, which permits a construction of the Fraunhofer diffraction pattern of light from any combination of slits. The diagram employs a graphical evaluation of a line integral over a frozen-wave profile along a rotated slit configuration.
The generalized raindrop problem
Authors: Hossein Partovi M. , Aston Duane R.
Published: March 2005

The problem of a raindrop falling through mist is generalized to include air resistance and change of shape. The resulting equation of motion is reduced to quadratures by standard methods. Although these models and their solutions are original and bear some relevance to the realistic problem of raindrop motion, the presentation is mainly pedagogical and intended to illustrate a wide range of concepts and methods in analytical mechanics. Accordingly, air resistance, mist accretion and change of shape, the use of the energy integral by analogy, terminal motions and terminal solutions, formulation by energy methods, degradation of the effect of drag in the presence of mist accretion, asymptotic expansions, and numerical results are discussed in some detail.
Using nonexponential absorption to demonstrate that anode x rays are polychromatic
Authors: Piccard Richard D. , Carter Sue A.
Published: March 2005

Refinements are reported to the technique suggested by TELTRON for student demonstration of the polychromatic nature of the x rays generated by a conventional x-ray tube, such as the TEL-X-OMETER apparatus, by observing nonexponential absorption in aluminum. Excluding a linear fit on the semilog plot of transmission versus absorber thickness requires a thorough error analysis, propagating the uncertainty estimates. A stored program is described, which accomplishes that analysis using the minitab statistics package.
Helmholtz coils revisited
Authors: Purcell Edward M.
Published: March 2005

Two circular coaxial current rings of radius a, separated by a distance b, are usually called Helmholtz coils if b=a. That is the well-known prescription for greatest uniformity of field in the neighborhood of the center of symmetry. Examined here is the magnetic field of a pair of current rings at a distance r from the center large compared to the ring radius a. Expanding the distant field in powers of a/r, each term may be associated with a magnetic 2k-pole source and a field falling off as (a/r)k+2. The dipole field k=1, falling as (a/r)3, has as its source the total dipole moment of the two rings. It eventually swamps the contribution of higher multipoles. The term (a/r)5 with k=3 has for its source the octupole (23-pole) moment of the pair of rings. Its coefficient vanishes if b=a. Thus the Helmholtz pair, thanks to its seemingly irrelevant specialization for uniform central field, is also endowed with zero octupole moment. The term-by-term connection between the expansion of the distant field and the expansion of the central field holds for currents entirely confined to the surface of a sphere, which is the case for the Helmholtz pair of thin current rings. The dipole field can be nullified by nesting the Helmholtz pair within a larger Helmholtz pair designed to have equal area × ampere–turns but carrying reversed current.

At the cost of a minor reduction in central field and without compromising the central field’s uniformity, the total dipole moment of the four rings can thus be made zero. The current system now has zero dipole moment and zero octupole moment. As 2k -poles with k even are ruled out by symmetry, the lowest surviving multipole is the 25-pole, the ‘‘32-pole.’’ The residual distant field, of which it is now the dominant source, falls as (a/r)7. An exact calculation of the ratio of field magnitude B to central field B0 shows that B/B0 is closely proportional to (a/r)7 everywhere in the field beyond r≊5a. Already at r≊6a, B/B0 has fallen to 10−4. It is shown how equivalent results can be achieved with coils of finite cross section. This scheme can drastically reduce the ‘‘stray field’’ around a superconducting magnet. In effect, it closely confines most of the return flux. Another application involves the induction of alternating currents by precessing magnetic moments. The four coils connected in series constitute a receiving coil tightly coupled to any oscillating dipole at its center, but virtually immune to disturbance by external sources.
Exact expressions for ferromagnetic magnetization in the mean field theory
Authors: Whitaker M. A. B.
Published: March 2005

The conventional analysis of the Weiss mean field theory of a ferromagnet uses a graphical treatment for the relationship between M, T, and the applied magnetic field, B0. It is shown that, for spin 1/2 and spin 1, exact expressions may be obtained for T(M,B0), and the behavior of the function is analyzed.
Fresnel diffraction by spherical obstacles
Authors: Hovenac Edward A.
Published: March 2005

Lommel functions were used to solve the Fresnel–Kirchhoff diffraction integral for the case of a spherical obstacle. Comparisons were made between Fresnel diffraction theory and Mie scattering theory. Fresnel theory is then compared to experimental data. Experiment and theory typically deviated from one another by less than 10%. A unique experimental setup using mercury spheres suspended in a viscous fluid significantly reduced optical noise. The major source of error was due to the Gaussian-shaped laser beam.