Edwin F. Taylor

When action is not least

We examine the nature of the stationary character of the Hamilton action S for a space-time trajectory (worldline) x(t) of a single particle moving in one dimension with a general time-dependent potential energy function U(x,t). We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials U(x), the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum, that is, it is never greater along the actual worldline than along every nearby alternative curve. We illustrate these results for the harmonic oscillator, two different nonlinear oscillators, and a scattering system. We also briefly discuss two-dimensional examples, the Maupertuis action, and newer action principles.

Teaching Feynman's sum-over-paths quantum theory

We outline an introduction to quantum mechanics based on the sum-over-paths method originated by Richard P. Feynman. Students use software with a graphics interface to model sums associated with multiple paths for photons and electrons, leading to the concepts of electron wavefunction, the propagator, bound states, and stationary states. Material in the first portion of this outline has been tried with an audience of high-school science teachers. These students were enthusiastic about the treatment, and we feel that it has promise for the education of physicists and other scientists, as well as for distribution to a wider audience.

Propagator for the simple harmonic oscillator

Through direct simulation with a computer program, we have verified that the published formula for the propagator of the one-dimensional simple harmonic oscillator (1DSHO), when applied in the straightforward manner in which physicists might interpret it, predicts the wrong development of the wave function for half the future times. This note presents a new and simple derivation of the (known) explicit phase factors of the normalization of the 1DSHO propagator that yield the correct development of any initial wave function for all times.

Teaching Physics On Line.

Since 1986 we have offered courses on special relativity and general relativity in a variety of formats over computer networks nationwide to a mix of high school students, high school teachers, college students, and college professors. Using ideas from many sources, we have evolved a highly structured format that keep students working every day and talking extensively with one another over the conference system. Obligatory student ‘‘Reading Memos’’ bring a flood of comments and complaints essential to evaluation of student progress and revision of the text. We consider possible reasons why these courses from the United States have not worked with participants in Europe, and speculate about possible consequences of future technological developments.

From conservation of energy to the principle of least action: A story line

We outline a story line that introduces Newtonian mechanics by employing conservation of energy to predict the motion of a particle in a one-dimensional potential. We show that incorporating constraints and constants of the motion into the energy expression allows us to analyze more complicated systems. A heuristic transition embeds kinetic and potential energy into the still more powerful principle of least action.

The Rise and Fall of PSI in Physics at MIT.

Small‐scale trials of the Personalized System of Instruction (PSI) in physics at MIT under the sponsorship of the Education Research Center were followed by adoption of large‐scale introductory PSI courses in the Department of Physics. After four semesters, all introductory physics PSI courses were suspended. This study investigates what happened. We point out organizational difficulties in going from small‐ to large‐scale use of the PSI method. We also identify certain conflicting perceptions among participants and observers that contributed to the suspension. The study, we hope, is an example of an informative ’’postmortem’’ of a type much needed in higher education.

Quantum physics explains Newton's laws of motion

Newton was obliged to give his laws of motion as fundamental axioms. But today we know that the quantum world is fundamental, and Newtons laws can be seen as consequences of fundamental quantum laws. This article traces this transition from fundamental quantum mechanics to derived classical mechanics.

Deriving Lagrange's equations using elementary calculus

We derive Lagrange’s equations of motion from the principle of least action using elementary calculus rather than the calculus of variations. We also demonstrate the conditions under which energy and momentum are constants of the motion.

Space‐time software: Computer graphics utilities in special relativity

No one can experience directly the world of the very fast, described by special relativity. Interactive graphics displays have been developed for personal computers that help students visualize this world. They are in the form of interactive graphics utilities that students use to carry out homework exercises and take‐home projects. Sequential versions of these programs have been used for 3 years in classes in various institutions. This article describes the programs and reports on the educational outcomes of these computer uses.

Limitation on proper length in special relativity

The proper length L* of an accelerating object is limited to the values L*≤c2/a*F where a*F is the proper acceleration of the front end. For the maximum length, the acceleration of the rear end is infinite. The effect rests on the relativity of simultaneity, one of the most difficult relativity concepts for students. We explore some consequences of this result.