Volume 64, Issue 12, December 1996
 Papers


The uncertainty principle for energy and time
View Description Hide DescriptionIt is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well‐known relation between the lifetime of a state and its energy spread.

Cosmological Schwarzschild radii and Newtonian gravitational theory
View Description Hide DescriptionWe describe Friedmann–Robertson–Walker zero‐pressure dust‐filled universes using a Schwarzschild‐like curvature spatial coordinate R along with the usual cosmological time coordinate t. In terms of coordinates (R,t), the geodesic equations of general relativity for the motion of the galaxies comprising the universe satisfy exactly a Newtonian inverse‐square relationship. This allows us to formulate relativistic cosmology by starting first with Newtonian cosmology. After the mathematics of Newtonian cosmology is worked out, the general‐relativistic metric for the universe is constructed from a well‐defined prescription, and the behavior of light signals is then determined from the metric. It is found that certain radial light signals born at the Big Bang eventually reach a maximum distance in their journey through the universe, where they turn around and return to some arbitrarily chosen origin. The turning around takes place at an apparent horizon located at a Schwarzschild radius in the universe.

Restoration of interference and the fallacy of delayed choice: Concerning an experiment proposed by Englert, Scully, and Walther
View Description Hide DescriptionA two‐slit experiment with atoms, proposed by Englert, Scully, and Walther, appears to permit experimenters to choose, after each atom has made its mark on the screen, whether the atom has passed through a particular slit or has, in some sense, passed through both of them. Through a misleading wording these authors even appear to endorse this interpretation. In actual fact, this choice exists only until the atom hits the screen. The said experiment thus is a ‘‘delayed‐choice’’ experiment only in the semantically contingent sense of Wheeler.

A geometric‐algebra treatment of the Feynman–Vernon–Hellwarth space in the two‐state problem
View Description Hide DescriptionPauli algebra is reviewed and then related to Dirac’s braket algebra and a generalization of the Wigner–Jordan operators in the two‐state problem. A method similar to the Feynman–Vernon–Hellwarth (FVH) method is developed from the Pauli algebra density operator. This yields a true geometric algebra treatment of the FVH space. The equation of motion for the density operator is developed from the Schrödinger equation. A general solution in laboratory coordinates is given. Ensemble effects are discussed.

A simple model for Faraday waves
View Description Hide DescriptionWe show that the linear‐stability analysis of the birth of Faraday waves on the surface of a fluid is simplified considerably when the fluid container is driven by a triangle wave form rather than by a sine wave. The calculation is simple enough to use in an undergraduate course on fluid dynamics or nonlinear dynamics. It is also an attractive starting point for a nonlinear analysis.

A simple model for nonequilibrium fluctuations in a fluid
View Description Hide DescriptionIn equilibrium systems, thermal fluctuations are correlated in time (fluctuation–dissipation theorem) yet their instantaneous values are uncorrelated in space over macroscopic distances. Out of equilibrium, long range spatial correlations are predicted by theory and have been observed in both laboratory experiments and computer simulations. In this paper, we present a simple model, called the train model, that illustrates this phenomenon. The theoretical analysis of the model and its connection with fluctuating hydrodynamics are outlined.

Using qualitative problem‐solving strategies to highlight the role of conceptual knowledge in solving problems
View Description Hide DescriptionWe report on the use of qualitative problem‐solving strategies in teaching an introductory, calculus‐based physics course as a means of highlighting the role played by conceptual knowledge in solving problems. We found that presenting strategies during lectures and in homework solutions provides an excellent opportunity to model for students the type of concept‐based, qualitative reasoning that is valued in our profession, and that student‐generated strategies serve a diagnostic function by providing instructors with insights on students’ conceptual understanding and reasoning. Finally, we found strategies to be effective pedagogical tools for helping students both to identify principles that could be applied to solve specific problems, as well as to recall the major principles covered in the course months after it was over.

Realization of an interaction‐free measurement of the presence of an object in a light beam
View Description Hide DescriptionThe gedanken experiment of Elitzur and Vaidman showing wave‐particle duality was realized as a demonstration at a scientific exposition. Presence or absence of a mirror in one arm of an interferometer was measured while detecting one photon at a time. A reasonable fraction of events was ‘‘interaction‐free,’’ which means that the conclusion that there was a mirror in the beam could be drawn from the signal of a photon that had not interacted with the mirror. The demonstration was found to be thought‐provoking.

Reflections on the pedagogic motive power of unconventional thermodynamic cycles
View Description Hide DescriptionPedagogic niceties in the treatment of unconventional thermodynamic cycles, especially those involving (negatively sloping) diagonal linear transitions in a P/V state diagram and/or those implying supposedly superefficient heat‐engine operation, are discussed as a means of stimulating student interest and comprehension, as well as promoting fresh insights, correcting erroneous notions, and provoking further enquiry. In particular, a novel (ostensibly all‐adiabatic) engine using two ideal gases of mutually differing atomicities as working substance is analyzed qualitatively and quantitatively. Emphasis is placed on the crucial role of the second law of thermodynamics in a determination of heat‐engine operation.

A pedagogical model of primordial helium synthesis
View Description Hide DescriptionThe calculation of the primordial hydrogen and helium abundances in the big‐bang cosmology is presented in an oversimplified model accessible to university physics students who have had no physics beyond an elementary modern physics course.
