Volume 55, Issue 1, January 1987
 Letters To The Editor



The Bloch wave vector and textbooks
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 Papers


Correlations in separated quantum systems: A consistent history analysis of the EPR problem
View Description Hide DescriptionThe familiar problem of two separated, noninteracting spin‐ 1/2 particles in a state of zero total spin is analyzed using the consistent history interpretation of quantum mechanics and shown to behave in many respects like a classical system of two noninteracting objects whose individual properties are unknown but strongly correlated with each other. There is no action at a distance between the particles and a measurement on one has no effect whatsoever on the other. However, the result of a measurement of a spin component of one of the particles can be used to infer (correctly) its value prior to the measurement, and also the corresponding spin component of the other particle at all times prior to when that particle interacts with something else. In these respects the quantum system behaves like its classical counterpart. On the other hand, the paradoxical (nonclassical) aspects of the quantum situation seem to be precisely those already present in the quantum theory of a single particle.

Electron trajectory in an e/m experiment
View Description Hide DescriptionA common undergraduate experiment to measure the e/m ratio uses a magnetic field produced by Helmholtz coils to deflect an electron beam. It is usually assumed that the beam is in the symmetrical plane of the coils and that it is in a uniform field identical to the field at the center of the coils. The magnetic induction field at the central position can be calculated from the geometry of the coils. In this paper the magnetic field in the region of the electron beam is calculated and compared to the field at the center of the coils. The height of the beam increases as it traverses the semicircular path and this increase is calculated.

In memoriam J. Jaumann: A direct demonstration of the drift velocity in metals
View Description Hide DescriptionMoving a Hall specimen perpendicular to the magnetic field and in the opposite direction to the drift motion of carriers with exactly the drift velocity results in a compensation of the Hall voltage. Thus a drift velocity of, e.g., 0.6 mm s^{−} ^{1} in Cu can be directly observed.

The early history of cosmic ray research
View Description Hide DescriptionWe review the prehistory and early history of cosmic ray studies, concentrating on the period 1900–1927. Following the discoveries of the electron and radioactivity just before the turn of the century, the old problem of leakage of charge from a conductor in air was investigated in terms of the new concept of ionization and ionizing radiation, part of which was found to be highly penetrating and to be of extraterrestrial origin. At first supposed to consist only of ultrahigh energy gamma rays, the cosmic ray primaries are now known to be mainly charged particles. The modern period of cosmic ray research began in 1927, when individual particles were studied by cloud‐chamber and coincidence counting techniques.

The vibrating string controversy
View Description Hide DescriptionIn the mid‐1700s a debate raged between Jean d’Alembert, Leonhard Euler, and Daniel Bernoulli concerning the proper solution to the classical wave equation. This controversy was partially solved by Lagrange and, more conclusively, by Fourier (50 years later) and it provides an interesting case study for the role of mathematics in the modeling of physical phenomena. Of particular note in this debate, was the meaning of boundary conditions. The controversy is summarized from the point of view of this mathematical physics perspective.

The fly ball trajectory: An older approach revisited
View Description Hide DescriptionAn older approach to the problem of projectile motion with quadratic drag force is presented with the fly ball as an example. In this approach, analytical solutions for the velocity, curvature, and arc length are obtained as functions of the slope angle. It is shown that the velocity and curvature do not have their extrema at the top of the trajectory but during the early phase of descent. The entire problem is reduced to simple integrations over the slope angle.

The lateral force on a spinning sphere: Aerodynamics of a curveball
View Description Hide DescriptionThe lateral force on a spinning baseball in a wind tunnel has been measured. The magnitude of the force is nearly independent of the orientation of the seams of the ball. The drag coefficient appears to be at most weakly dependent on Reynolds number and to be principally a function of the ratio of the rotational speed of the equator of the ball to the wind tunnel speed. This is to be compared to the work of Briggs, which implies a strong effect of Reynolds number on the drag coefficient.

The dynamic shear modulus and internal friction of a fiber vibrating in the torsional mode
View Description Hide DescriptionA torsional pendulum consisting of an 8‐mil boron–tungsten fiber and inertial member is described. The torsional oscillations are excited electrostatically and detected by a frequency modulation technique. Both the dynamic shear modulus and the internal friction can be measured. In addition, this system exhibits the resonance behavior characteristic of a high Qoscillator.

A computer‐managed undergraduate physics laboratory
View Description Hide DescriptionSeventeen one‐semester undergraduate laboratory courses are managed by a microcomputer system at Concordia University. Students may perform experiments at any time during operating hours. The computer administers pre‐ and post‐tests. Considerable savings in manpower costs is achieved. The system also provides many pedagogical advantages.

A rotary Curie point magnetic engine: A simple demonstration of a Carnot‐cycle device
View Description Hide DescriptionThe fact that metallic gadolinium is ferromagnetic with a Curie temperature of 294 K is used in the constuction of an inexpensive and simple thermomagnetic engine, which can be driven by an electric lamp or by sunlight. The apparatus can be used as a simple demonstration of the Carnot principle.

A comparison of explanations of the aerodynamic lifting force
View Description Hide DescriptionThe frequently used explanation of the aerodynamic lifting force using Bernoulli’s law is shown to have fundamental drawbacks and incomplete reasoning. On the other hand, an explanation based on repulsion forces is shown to be complete, easier to assimilate, more powerful, and better related to the basic principles of mechanics. Furthermore, it is shown that the high streaming velocity at the upper side of the aerofoil is not the reason for the low pressure. To the contrary, the low pressure generated by the aerofoil is the reason for the high streaming velocity.

Bound states in weak attractive potentials in one‐dimensional quantum mechanics
View Description Hide DescriptionIn one‐dimensional quantum mechanics, there is one bound state near zero energy in a weak attractive potential. A series is derived for the bound state energy in powers of the potential. Normally the leading term is O[V ^{2}], but when the potential has a long attractive tail, −V _{0}‖a/x‖^{ P }, 0<p<1, the leading term is O[(V _{0})^{2/(2−p)} ].

Symbolic proof of the Helmholtz theorem
View Description Hide DescriptionSymbolic proof of the Helmholtz theorem on the unique determination of a vector field from its divergence and curl is given which requires only an elementary knowledge of electrostatics and vector analysis. The conditions for uniqueness emerge as natural consequences of the symbolic method employed.

Determination of gravitational acceleration using a rubber ball
View Description Hide DescriptionA rubber ball, with a 4.9‐cm diameter and 64‐g mass, is dropped from a certain height h to bounce onto the floor, and the time between the first bounce and the nth bounce is measured. In the first version this time is taken by a digital stopwatch with one centisecond resolution, and in the second version, by an integrated‐circuit timer with 1‐ms resolution, triggered by a microphone. Considering the formula h=(1/2)g t ^{2} and the coefficient of restitution of energy in the collision of the ball against the floor, the acceleration of gravity g may be determined to 2%–3% in the first version and to better than 1% in the second one. The experiment opens many problems, i.e., about the time of contact, the internal vibrational energy of the ball, and the dependence of the elastic properties of the rubber on the temperature.

Newton’s law of motion for variable mass systems applied to capillarity
View Description Hide DescriptionThe standard Newton’s equation of motion for variable masses is set up and solved to describe the time‐dependent rise of a liquid in a capillary tube. The findings in the nonviscous case are also supported by quantitative estimates of the potential energy changes suffered by the interfacial films. The theoretical oscillations in the viscous case are confirmed experimentally for tubes having a radius larger than a critical value.

A simple geometric model for visualizing the motion of a Foucault pendulum
View Description Hide DescriptionA geometrical model of the Foucault pendulum is presented, which corrects some common misconceptions concerning the ‘‘fixed’’ plane in which the pendulum oscillates. It is shown visually, and by the use of metric geometry, that the direction of oscillation of the pendulum undergoes a parallel displacement as it moves along the Earth’s spherical surface. However, this direction of oscillation is not fixed relative to the stars.

A relativistic mass tensor with geometric interpretation
View Description Hide DescriptionWe derive a relativistic mass tensor (dyadic or matrix) whose origin and properties have a direct geometric interpretation in terms of projection operators related to the particle’s world line and local inertial frame in Minkowski space, yet whose eigenvalues are simply the longitudinal (m _{ l }) and the transverse (m _{ t }) mass. Writing the noncovariant equations of motion (EOM) for a point particle in terms of this mass tensor bridges the gap between the compact but sterile form of the Lorentz covariant EOM and the usual (‘‘unwieldy’’) noncovariant EOM in which m _{ l } and m _{ t } appear. General expressions for 3‐ and 4‐space mass (inverse mass) tensors are presented in terms of the system Lagrangian (Hamiltonian).

The depolarization field inside a homogeneous dielectric: A new approach
View Description Hide DescriptionA simple argument is presented to make plausible the theorem that a homogeneous dielectric placed in a uniform external field is uniformly polarized only if its shape is ellipsoidal. Expressions for the depolarization factors are displayed in a form which makes immediately apparent the well‐known sum rule. An additional symmetry relation between the depolarization factors is used to write integral expressions for them. It is shown that these factors reduce to the standard form. Neither Dirichlet’s integral representation nor ellipsoidal coordinates are used in this derivation.
