Pauli equation and the spin

Pedro Pereyra Padilla

Alexander Komech

Espen Gaarder Haug

In this short note we will show the Haug-1 quantum wave equation in relation to where we incorporate spin following the Pauli route. This lead to a similar equation to the Schrodinger-Pauli equation, but our equation is relativistic while the Schrodinger-Pauli equation is non-relativistic, our equation is also simpler in terms of for example the time and space are on the same order (first derivatives), while in the Schrodinger-Pauli equation time is on first order and spatial-spin dimensions on second order. Comments are welcome.

James M Wilkes

Abstract We review the literature on the Pauli equation and its current density, discussing the progression from the original phenomenological version of Pauli to its derivation by Lévy-Leblond from a linearization of the Schrödinger equation. It was established conclusively by Lévy-Leblond’s work that the spin of a spin-1/2 particle such as an electron is non-relativistic in nature, contrary to what was often stated following Dirac’s derivation of a relativistic wave equation, and his subsequent demonstration that Pauli’s spin interaction term appeared in the non-relativistic limit. In this limit, the Gordon decomposition of the associated probability current density was found to contain a spin-dependent term. Such a term does not follow, however, from the usual derivation of the current density from the Pauli equation, although various physically motivated but otherwise ad hoc explanations were put forward to account for it. We comment on the only exception to these of which we are aware implying the spin term in the current was in fact non-relativistic in nature. However, the earlier work of Lévy-Leblond had already shown, with no additional assumptions, that this term was a prominent feature of the current density derived from his equation. Hence, just as with the spin itself, the spin current was non-relativistic, claims to the contrary notwithstanding. We present a somewhat simplified derivation of the Lévy-Leblond equation and its current density, commenting on possibilities for experimental work that might indicate measurable consequences of the spin term in the current density. The presentation is at a level of quantum mechanics, electromagnetic theory, and mathematical methods typical of courses taken by first-year graduate students in physics, assuming some familiarity with Pauli spin matrices and the Dirac equation. Calculations are presented in detail, making the material suitable for use by teachers as supplementary problems for assignments, and development of theoretical concepts peculiar to the Pauli equation and its probability current density.

Jędrzej Śniatycki

Ronald J. Alder, Robert A. Martin

The gyromagnetic ratio or g factor of the electron is correctly predicted by the Dirac equation to be 2. However, it is also possible to obtain this value from nonrelativistic quantum mechanics, as shown by Sakurai, who credits Feynman. It is noted here that the same trick used by Sakurai allows a factorization of the time independent Pauli equation. The result is an equivalent first-order equation which is referred to as the spin-momentum equation. In the absence of a scalar potential, the spin-momentum equation is the eigenvalue equation for the inner product of spin and kinetic momentum operators. The spin-momentum equation is solved directly for a constant magnetic field, and the solution is compared with the well-known Pauli equation solutions.