Harry M. Schey’s research while affiliated with Massachusetts Institute of Technology and other places

The paper describes the designing and testing of a laboratory, computer and calculus based course in mathematics. The laboratory is central to the course and stimulates in the student the need and desire to know more about mathematics. Further, it enables mathematics to be taught in a real world context. Computers are used to take the drudgery out of the mathematics and make it possible to attack real scientific and technical problems. This new approach to calculus is less formal and depends to a smaller extent upon prior mathematical training so that it appeals to a much wider audience. The proposed course, with its emphasis on laboratory measurements, is ideally suited to the exploration of numerical methods and their application to the calculation of derivatives, definite integrals and the solution of differential equations.

We investigate the headlight effect for nonelectromagnetic radiation and find that, depending upon the wave and source velocities, the effect may be normal (radiation concentrated in the forward direction) or anomalous (radiation concentrated in some direction other than forward). Expressions are derived for the distribution of the radiation as a function of angle for both an infinite line source and a point source, and the critical wave velocity at which the headlight effect makes a transition from anomalous to normal is determined for both cases as a function of source velocity. A possible application to the decay of radioactive particles is suggested.

We describe a computer-generated film which shows an expanding wavefront as seen from a coordinate frame relative to which the source moves with constant velocity. An expression for the wavefront as a function of time is derived, and features of it are discussed in terms of the relativistic laws for the addition of velocities. Sample stills from the film are presented and used in discussing the shape of the wavefront, the velocity of the center of the wavefront, and the headlight effect, all as functions of the signal velocity and the velocity of the source.

We describe the details involved in presenting the time development of one-dimensional quantum-mechanical systems in the form of computer-generated motion pictures intended for pedagogic purposes. Concentrating on reflection-transmission phenomena, we formulate the problem in terms of a Gaussian wave packet impinging on a square well or barrier and being reflected and transmitted. The wave equation is solved numerically by methods discussed in detail and photographs of the wave packet vs position at a variety of times and for a range of projectile energies are given.

A method of counting bound states is modified and applied to singular helium intermolecular potentials to determine the conditions under which such potentials support a single bound state. Results are presented for the "6-9" and "6-12" potentials and compared with the results of earlier variational calculations.

We propose a classification scheme for jj-coupled nuclear-shell-model wave functions which is based on spatial-symmetry considerations. The wave functions are determined by diagonalizing the Majorana operator, each wave function being labeled by the corresponding eigenvalue. We assume that among the states with quantum numbers jnTJ the one with the largest Majorana eigenvalue lies lowest in energy, and these maximum-symmetry wave functions are used in calculations of beta-decay transitions and magnetic moments for nuclei in the 1f72 shell. The beta-decay results are considerably better than those obtained with seniority wave functions or with the wave functions derived by Lawson using Elliott's generating procedure. They also compare very favorably with the results of McCullen, Bayman, and Zamick based on an interaction calculation which uses an empirically determined two-body potential. The magnetic-moment calculations are somewhat less successful; except for the seniority scheme, the other theories usually give better results.

For the case of a short-range central potential, the quantity alphal, defined as the zero-energy limit of k2l+1deltal, vanishes whenever the range and depth of the potential are such that there is a state of zero binding energy. By solving the zero-energy scattering problem we obtain alphal as a function of range and depth and thus determine the number of bound states supportable by a given central potential as a function of the potential parameters without having to solve the associated and more difficult eigenvalue problem. The method is applied to the Debye-Hückel (Yukawa) and Woods-Saxon potentials.

Quantum corrections to the Thomas-Fermi model of the atom derived by expansion procedures are examined numerically. Corrected potentials, one-electron eigenvalues and eigenfunctions, dipole oscillator strengths, diamagnetic susceptibilities, and electron binding as a function of atomic number are considered. In all cases the corrected quantities either agree less well with experiment than the results of the simple zeroth-order theory or show improvement which is inconsistent, unimpressive, and unreliable. These results lend strong support to conjectures made by others that the expansion methods leading to the quantum corrections give rise to series which are asymptotic and which fail to include important oscillatory terms.