Generalization of the Schrödinger Theory of Electrons

Viraht Sahni

Viraht Sahni

Viraht Sahni

A.P. French, Edwin F. Taylor

There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.

S. Rupp

In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.

David Hestenes, Richard Warren Gurtler

The Pauli theory of electrons is formulated in the language of multivector calculus. The advantages of this approach are demonstrated in an analysis of local observables. Planck's constant is shown to enter the theory only through the magnitude of the spin. Further, it is shown that, when obtained as a limiting case of the Pauli theory, the Schrödinger theory describes a particle with constant local spin. An important consequence of this result is the realization that the usual interpretations of the Dirac and Schrödinger theories are mutually inconsistent in certain details.