Volume 64, Issue 3, March 1996
 Papers


Resource Letter CD‐1: Causality and determinism in physics
View Description Hide DescriptionThis Resource Letter provides a guide to the literature on causality and determinism in physics. It first surveys the literature of a long history of attempts to construct a systematic account of causality, attempts that constitute options for the use of the concept in physics. It then surveys the status of determinism in classical, relativistic, and quantum theories.

The pumping of a swing from the standing position
View Description Hide DescriptionThe pumping of a swing from a standing position is modeled as a rigid object forced to rotate back and forth at the lower ends of supporting ropes. This model after some approximations leads to a harmonic oscillator with driving and parametric terms. It is then argued that in the regime of the common playground swing the driving terms dominate and the pumping of a swing in the standing position is best characterized as a driven oscillator. Examination of the relative phase of the swinger and the swing also supports this conclusion. This model is compared with earlier work which claimed that the swing pumped by a standing swinger is characterized as a parametric oscillator. Simple demonstrations of both mechanisms are described. A comparison of pumping while standing and seated is also made.

Measuring the speed of digital signals (speed of light)
View Description Hide DescriptionAn odd number of digital not‐gates connected in a closed loop circuit will produce a relatively stable free running oscillator. The frequency of oscillation is determined by the number and type of not‐gates, the circuit supply voltage, and the total length of connection wire used between the not‐gates. Measuring the frequency for various lengths of connection wire enables the speed of the digital signals in the wire to be calculated. The method is sensitive enough to reveal that signals traveling in 22 gauge insulated copper wire propagate about 5% slower than those in 22 gauge bare copper wire. The speed in insulated wire is (2.87±0.03)×10^{8} m/s and the speed in bare wire is (3.03±0.02)×10^{8} m/s. This experiment has been found to be a very popular undergraduate laboratory due to its low cost, straightforward theory, and relatively high (∼1%) precision. The absence of a light source eliminates complicated optics and impresses students with the fact that visible light is not the only thing that travels with the speed of light!

The mathematics of Brownian motion and Johnson noise
View Description Hide DescriptionOne reason why Brownian motion and Johnson noise are difficult subjects to teach is that their mathematical requirements transcend the capabilities of ordinary differential calculus. Presented here is an exposition of the needed generalization of calculus, namely continuous Markov process theory, in a form that should be accessible to advanced physics undergraduates. It is shown how this mathematical framework enables one to give clear, concise derivations of all the principal results of Brownian motion and Johnson noise, including fluctuation–dissipation formulas, auto‐covariance transport formulas, spectral density formulas, Nyquist’s formula, the notions of white and 1/f ^{2}noise, and an accurate numerical simulation algorithm. An added benefit of this exposition is a clearer view of the mathematical connection between the two very different approaches to Brownian motion taken by Einstein and Langevin in their pioneering papers of 1905 and 1908.

Photon counting statistics—Undergraduate experiment
View Description Hide DescriptionA photon countingexperiment for student physics laboratory is described. It is designed to illustrate the probabilistic nature of the photodetection process itself as well as statistical fluctuations of light. The setup enables the student to measure photon count distributions for both coherent and pseudothermal light sources yielding Poisson and Bose–Einstein distributions, respectively.

Photon states made easy: A computational approach to quantum radiation theory
View Description Hide DescriptionStudents first meet the wave‐particle paradox through the photon and wave descriptions of light. Yet, in basic courses on quantum mechanics, they study matter particles only, because the mathematics of the quantized radiation field is usually considered too advanced. An oscillatingelectromagnetic field is formally similar to a harmonic oscillator, whose energy eigenstates can represent states of well‐defined photon number. Using a computer program from the CUPS project, an approach will be described which demonstrates the action of the annihilation operator on these states, constructs coherent states which behave like classical electromagnetic fields, and shows how such states can be squeezed. All of these have practical relevance in modern optics. This is just one example of the computer making a hitherto unapproachable subject accessible to ordinary undergraduates.Computers have already changed how much of quantum mechanics is taught. As more such possibilities are realized, the teaching of the whole subject must surely change radically.

Using a linear approximation for single‐surface refraction to explain some virtual image phenomena
View Description Hide DescriptionThere are many examples of interesting optical phenomena involving virtual images arising from refraction at the interface(s) between air and some transparent solid or liquid. However, because it is cumbersome to interpret analytical expressions or to sketch diagrams using Snell’slaw of refraction, students rarely explore and develop a qualitative understanding of these phenomena. In this paper we introduce a simple‐to‐use linear approximation for single‐surface refraction and show how it leads to qualitatively correct descriptions of some interesting but complex optical phenomena. In particular, we analyze the virtual images formed when looking at objects through both rectangular blocks and triangular prisms. The refraction images observed through a triangular prism are particularly interesting pedagogically and also provide a physical means to distinguish between objects located just outside and just inside, or on, the prism’s surface.

A time‐dependent study of resonant tunneling through a double barrier
View Description Hide DescriptionThe resonant tunneling of a wave packet is studied, for a one‐dimensional double barrier, by solving the time‐dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.

Viscous and nonviscous models of the partially filled rolling can
View Description Hide DescriptionWe examine the motion of soft drink cans rolling down an incline as a function of the amount of water in the can. Our observations show this behavior to be remarkably complex. We introduce models suggested by the experimental results which describe the limiting cases of a nonviscous fluid and an infinitely viscous fluid, and compare the results of the model calculations to our observations.

Low‐cost differential scanning calorimeter
View Description Hide DescriptionWe present a simple and inexpensive calorimetric system that has been implemented in the undergraduate students laboratory of thermodynamics at the Physics Faculty of the University of Barcelona. It is shown that, after proper calibration, the system enables measurement of the relevant thermodynamic quantities at a first‐order phase transition. As an example, the solid–liquid phase transition of water can be studied: Students find that both the change in specific heat and the latent heat (and its temperature dependence) coincide within the experimental scatter with the values given in the literature.

Linear, one‐dimensional models of the surface and internal standing waves for a long and narrow lake
View Description Hide DescriptionLinear, one‐dimensional, analytical, and numerical models for seiches (standing waves) existing at the surface and on the internal interface between the upper warm and bottom cold layers for a long, narrow lake are developed. Using the specific bathymetric and thermal structures of a lake,Lake Champlain, these models predict the periods, nodal locations, vertical displacements, and current velocities associated with the fundamental and higher‐order modes. Modeled results are compared with historical limnological data obtained from Lake Champlain. Agreement is good for the periods of the first few modes of the surface and internal seiches; however, predicted current velocities associated with the internal seiche are about half as large as those actually observed.

Direct calculation of the electric field induced by a time‐varying magnetic induction
View Description Hide DescriptionThis paper deals with the problem of obtaining the electric field near a region of magnetic flux, whose geometry has cylindrical symmetry. We derive an expression for a direct calculation of the induced electric field vector in geometric situations where a slowly varying magnetic field is confined to a long solenoid of arbitrary cross section. We check our result in a standard case, then we use it in a nontrivial case to confirm its utility. The developed expression can also be used to provide numerical evaluation of the field within the class of cylindrical symmetry.

Toy calculations to estimate critical exponents for random walks
View Description Hide DescriptionA simple method based on the enumeration of all random walk configurations for a very reduced number of steps is used to obtain the critical exponent ν for self‐avoiding random walks in two and three dimensions.

On the bound states of a charged, nonrelativistic particle in the field of a stationary magnetic dipole: An introduction
View Description Hide DescriptionThe motion of ions above the earth’s atmosphere is introduced by discussing the nonrelativistic motion of a charged particle in the field of a stationary magnetic dipole. Unlike the Kepler or Coulomb problem, no way is known to separate our problem into one‐dimensional problems and a fortiori an exact analytic solution is not known. However, an effective potential can be identified whose study clarifies the qualitative and semiquantitative features of the bound states. We ask about the periodicity of bound states, taken for granted by one exposed only to the Kepler or Coulomb problem, and show how consideration of phase space can provide a partial answer, Poincaré recurrence. The treatment is accessible to students who have a first acquaintance with the canonical formalism and Liouville’s theorem, for example first‐year graduate students.

Use of pressure transducers in laboratory experiments. I. Experimental verification of the fundamental equation of fluid statics. Application to density measurements
View Description Hide DescriptionThe use of inexpensive pressure sensors, with minimum installation problems, has allowed several experiments to be designed for the physics class. An experimental device provided with a pressure sensor and a level sensor has now made it possible to verify the fundamental equation of fluid statics. The device has been applied to density measurements of several fluids (water, aqueous solution of NaCl and ethanol) at constant temperature.

Use of pressure transducers in laboratory experiments. II. Experimental verification of the Hagen–Poiseuille law. Application to viscosity measurement. Electrical analogy
View Description Hide DescriptionAs a continuation of part I of this work, further experiments were designed for the physics class using pressuresensors. An experimental setup using a pressuresensor and two optical fibersensors makes it possible to verify the Hagen–Poiseuille equation, to measure the viscosity of several gases and, by analogy with an electric current circuit, to verify that the parallel and series association of flow resistance may be solved as if they were electrical resistances.

Solving one‐dimensional Schrödinger‐like equations using a numerical matrix method
View Description Hide DescriptionA numerical method for the solution of one‐dimensional Schrödinger‐like equations with arbitrary numerical or analytical potentials is presented. The method takes advantage of matrix algebra for both obtaining several eigenvalues and eigenvectors at the same time and saving computer time. On the other hand, the method illustrates the close relationship between differential and algebraic eigenvalue problems, as well as the mathematical origin of quantization. Several examples are worked out in the text and the procedure for applying a user friendly routine to other problems is given.

The reciprocal lattice as the Fourier transform of the direct lattice
View Description Hide DescriptionWe argue that introducing the reciprocal lattice as the Fourier transform of the Bravais lattice provides not only a concise definition for the reciprocal lattice, but also makes it a concept easy to grasp. To illustrate the power of the concept we derive the reciprocal lattices of the bcc and fcc structures directly from their conventional unit cells, rather than from the primitive unit cells.

An undergraduate experiment on x‐ray spectra and Moseley’s law using a scanning electron microscope
View Description Hide DescriptionWe describe an undergraduatelaboratory experiment developed for sophomore modern physics students, using the x‐ray analysis feature of a scanning electron microscope. The experiment described is a fundamental physics measurement performed with a state‐of‐the‐art apparatus not usually used in pedagogical physicsexperiments. The characteristic x‐ray spectrum of elements shows a simple progression that can be quantitatively explained by the conceptually important Bohr–Rutherford shell model of atomic structure. Students measure the Kα and Kβ x‐ray spectra from metals in the range of 12≤Z≤41. The measurements allow the verification of Moseley’s law scaling of x‐ray energies with atomic number and a quantitative measure of the data’s fit to the shell model.

The libration limits of the elastic pendulum
View Description Hide DescriptionThe motion of the elastic pendulum is ruled by coupling between the pendulum mode and the harmonic oscillator mode. The bob of the pendulum executes trajectories in a domain of the vertical plane, the shape and size of which are set by the initial conditions and the total energy of the pendulum. The limits of this libration, in exact resonance, are studied using parabolic coordinates. Other Hamiltonians, with similar interaction terms, exhibit similar libration limits.
