Charles Schwartz

Calculations in schrödinger perturbation theory

Abstract The evaluation of second- and higher-order perturbations of the energy by iterative solution of Schrodingers equation, rather than by evaluation of the well-known matrix formulas, is described and exploited. Several examples are worked out exactly for the hydrogen atom, to point the way for other more practical, but more involved, problems.

Numerical integration of analytic functions

A method of numerical integration is presented which is simple, versatile and accurate. It is particularly valuable for integrals with complicated end point singularities.

Variational calculations of scattering

Abstract It has been found that the behavior of calculated results using variational principles for scattering problems is rather different from that previously known for bound-state problems. As more and more adjustable terms are added to the trial function, the “stationary” value for the phase shift does not converge smoothly, but may on occasion turn out to be grossly inaccurate. In this paper the phenomenon is displayed and partially analyzed, but not completely understood. We seem to be stuck with the conclusion that for a given amount of computational labor, a scattering phase shift may be determined only to an accuracy an order of magnitude worse than that of the analogous eigenvalue problem.

New calculation of the numerical value of the lamb shift

Abstract The calculation of the second-order perturbation in hydrogen which gives the low-energy part of the Lamb Shift is attacked from a new approach, described in the preceding papers and here extended. A formula is gotten for ln(k0) involving a double integral. The final numerical evaluation using an electronic computer to sum the series expansion of this formula yields what we believe to be the most accurate value of ln(k0) yet given. Small discrepancies with the earlier results of Harriman are outside the realm of current physical significance, but do indicate that the reliability of the earlier results was badly overestimated. Approximate formulas for the radiative-perturbed wave functions are given; these may be quite useful for further calculations.

Asymptotic Search for Ground States of SU(2) Matrix Theory

We introduce a complete set of gauge-invariant variables and a generalized Born–Oppenheimer formulation to search for normalizable zero-energy asymptotic solutions of the Schrodinger equation of SU(2) matrix theory. The asymptotic method gives only ground state candidates, which must be further tested for global stability. Our results include a set of such ground state candidates, including one state which is a singlet under spin (9).

High‐accuracy approximation techniques for analytic functions

A generalization of the familiar mesh point technique for numerical approximation of functions is presented. High accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points. In particular, derivative operators are represented by highly nonlocal matrices; but this is no drawback when one has computing machines to perform the algebraic manipulations. Some examples are given from familiar quantum mechanical problems.

Uses of approximate wave functions

Abstract A procedure is given for using approximate wave functions (derived, say, by the Ritz variational method), to calculate properties of the system other than the energy to an accuracy much greater than that previously thought to be possible. The method is based on the form of perturbation theory discussed in the previous paper but stands by itself as an independent and, it is believed, very powerful innovation. Several sample problems, based on the two-electron atom, are worked out; and a program of greatly increasing the accuracy in the calculation of many properties of atomic systems is envisaged.

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Relativistic quaternionic wave equation

We study a one-component quaternionic wave equation which is relativistically covariant. Bilinear forms include a conserved four-vector current and an antisymmetric second rank tensor. Waves propagate within the light cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for two-particle wave functions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energy-momentum tensor.

Calculus with a quaternionic variable

Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn in trying to extend our reach to include quaternions. The noncommutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus, but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+δ) is a compact formula involving both F′(x) and [F(x)−F(x∗)]/(x−x∗). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.