Volume 47, Issue 9, September 1979
 Editorial
 Papers


Perception of perspective
View Description Hide DescriptionIt is shown that perspective distortion in photography can be explained analytically if the eye operates as a simple lens and the brain remembers the hyperbolic relationship between image size and distance.

Valid approximations of the dispersion equation
View Description Hide DescriptionThe problem of deriving accurate long‐wavelength approximations of a particular dispersion equation for electromagnetic waves in a magnetized plasma is examined. Because the dispersion equation includes dependence upon external parameters, the approximation obtained by straightforward expansion is invalid in certain cases. An accurate approximation is derived in these cases by a modified expansion technique. Although only one specific practical problem is studied here, this example involves and exhibits general techniques that have broad application in related problems. Some features of long‐wavelength low‐frequency waves in a magnetized high‐temperature plasma are illustrated.

Spreading of wave packets in the coordinate representation
View Description Hide DescriptionBy dealing with the Schrödinger equation for a free particle in the coordinate representation alone, the expression for ψ (r,t) at time t is derived in terms of ψ (r,t _{0}) at an initial instant t _{0} with the use of the ’’method of self‐similar solution of partial differential equations.’’ The use of Green’s function is necessary due to the complex nature of the Schrödinger equation. The same technique can also be applied to the case of spreading of wave packets in a dissipative medium.

Simple soliton solutions
View Description Hide DescriptionSolutions to linear wave equations can always be expressed as a superposition of special solutions called plane waves. Some solutions to an important class of nonlinear wave equations can be thought of as a superposition of special solutions called solitons. In this paper, we present a simple method for generating these special soliton solutions.

The physics of electrochromism: An advanced laboratory experiment
View Description Hide DescriptionAmorphous tungsten trioxide films offer an interesting electrochromic system whose properties are readily amenable to study in a senior undergraduate laboratory. Basic optical color centerproperties, and electron and proton diffusion processes are characterized for the H_{ x }WO_{3} produced; additional experiments including the variation of the chemical potential of the hydrogen component and low‐temperature exploration of the metal‐insulator transition are proposed for the advanced student. The samples are easily prepared and measured by elementary means. The broad solid‐state concepts involved enrich standard advanced laboratory repertoires of more conventional nuclear and NMRexperiments.

Discharge time for a capacitor
View Description Hide DescriptionThe complete discharge of a capacitor through a real resistor is limited by thermal noise. A meaningful discharge time t _{ d } is defined by equating the charge across the capacitor in a noiseless R C circuit to the rms charge fluctuation in a noisy circuit Q _{0}(t _{ d }) =[〈 (δQ)^{2}〉_{ t }]^{1/2}. Three derivations are reviewed for the well‐known result 〈 (δQ)^{2}〉_{ t }=k T C. One derivation shows explicitly that δQ (t) can be determined with the noise current i _{ n }(t) as its source. Several examples of potential interest in the classroom are discussed.

Teaching T h e T a o o f P h y s i c s
View Description Hide DescriptionA course in modern physics and Eastern mystical philosophy is described.

Correlation between logical abilities and success in physics
View Description Hide DescriptionSixty students in a second semester freshman physics class were administered a test of formal operational reasoning. The composite logical test score was correlated with students’ final examination grades and yielded a correlation of 0.49. Furthermore, the composite correlation distribution provided evidence that the ability to employ formal operational reasoning is a necessary, though not sufficient, condition for student success in physics. These results point to the importance of seriously considering including, within the framework of physics courses, interactions and experiences designed to develop logical reasoning.

Excited levels 1s n, n−1 of the two‐electron atom
View Description Hide DescriptionClosed formulas for the Coulomb and exchange integrals in the excited states 1s n l, l=n−1, of the two‐electron atom are applied to the computation of the ionization energy of the singlet and tripet 1s2p states from He i up to O vii and to singlet and triplet 1s3d and 1s4f states of He i and Li ii. Two methods are compared: (a) the use of first‐order perturbation theory considering the mutual electrrostatic repulsion e ^{2}/r _{12} as a small perturbation and (b) imposing as solution of the Schrödinger equation a symmetrized product of a Z and a screened Z−1 hydrogenic wave functions. Computed and experimental values of the ionization energy and of the singlet‐triplet interval are compared and the evolution of agreement with increasing Z and n is discussed.

Molecular beams and effusive flows
View Description Hide DescriptionMolecular flux emanating from an aperture in a beam oven and incident on a point on the beam axis is discussed. The vector flux density is characterized by its vector‐velocity (rather than speed) probability distribution. As a consequence, the integrals of the velocity components orthogonal to the beam axis yield the observed dependences of beam intensity upon aperture area and distance from the aperture. Both rectangular and circular apertures are considered. The velocity integrals for the circular aperture are exact and demonstrate how the velocity distribution of the flux density gradually changes from that of the Maxwell flux at the aperture to that of a beam flux at large distances from the aperture. (This cannot be shown with the speed distribution.) An expression then is obtained for the flux from a circular aperture incident on an off‐axis point by making use of the concept of a virtual aperture which simplifies the problem considerably. This expression then is used to calculate both the normal flux density distribution and the total flux incident on a finite plane collector. These treatments may be applied to systems with apertures and collectors of almost any shape and/or size.

The Zitterbewegung of a free localized Dirac particle
View Description Hide DescriptionMuch of the lore of the Zitterbewegung of a free Dirac particle is found to need revision for the situation when the particle is initially localized. We find that in this case the rapid violent oscillations of angular frequency ∠2m c ^{2}/h/ in the average position of the particle are a transient rather than sustained phenomenon. Instead, the Zitterbewegung is manifested by a growth in the size of the particle but this growth is many orders of magnitude smaller than h//m c, the magnitude usually quoted. This behavior is contrasted with that of the Foldy‐Wouthuysen position for which the Zitterbewegung is absent. Finally, the applicability of these ideas to the situation of a localized Dirac particle in an external field is commented on.

Application of the Dirac delta function to electric current and magnetic multipole distributions
View Description Hide DescriptionWe discuss the equivalent current density for different types of magnetic distributions, discrete as well as continuous, generated by electric currents. The equivalent current density for the point magnetic dipole and quadrupole is constructed from the basic current element. It is shown that the third‐order tensor representing the quadrupole moment satisfies certain cyclic relationships which are a consequence of current conservation. Surface and volume distributions of dipoles are briefly discussed and results analogous to the electrostatic case are given. Finally, starting from Poisson’s equation for magnetism, we directly obtain a multipole expansion of an arbitrary current distribution yielding the equivalent current density for each multipole.

The Slinky as a model for transverse waves in a tenuous plasma
View Description Hide DescriptionTransverse waves in a Slinky suspended by an array of strings share the dispersion relation for transverse electromagnetic waves in a tenuous plasma. Experimental results for the Slinky system are presented, suggesting a useful and simple laboratory exercise for an elementary physics course.

Eigenvalues of J ^{2}
View Description Hide DescriptionMost derivations of the possible eigenvalues of J ^{2} [i.e., h/^{2} j (j + 1), j = 0, 1/2, 1, 3/2,...] rely upon the eigenvalues, h/m, of a component of J, say J _{ z }. A derivation of the J ^{2}eigenvalues is presented in which the concept of J _{ z }eigenvalues does not appear. The derivation is based upon addition of angular momenta and the algebraic properties of the Pauli spin matrices.

Momentum operators for curvilinear coordinate systems
View Description Hide DescriptionExpressions for the covariant and contravariant components of the quantum momentum operator (h//i) ∇ are formulated for curvilinear coordinates in three‐dimensional space. The classical formalism is modified to take account of the noncommutivity of coordinates and momenta in the quantum case. These components are shown to be simply related to canonical momentum operators conjugate to the curvilinear coordinates. Expressions for components of angular momenta are also obtained. These results are specialized for orthogonal curvilinear coordinates and illustrated for spherical polar coordinates.

Dirichlet’s integral formula and the evaluation of the phase volume
View Description Hide DescriptionThe phase volume in the microcanonical ensemble is not easily obtained even in the cases where the Hamiltonian function is separable. Dirichlet’s integral formula constitutes a consistent method to evaluate the phase volume for simple situations. Several examples where Dirichlet’s integral formula is used to obtain the phase volume are presented in this paper.
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 Notes and Discussions


Group properties of quantities and units
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Rebuttal to de Boer’s ’’Group properties of quantities and units’’
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Observation of the first overtone vibrational mode in an automobile whip antenna
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