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Advanced Studies in Theoretical Physics

Generalized Levy-Leblond and Schrodinger equations for spinor wavefunctions

R. Huegele, Z. E. Musielak, John L. Fry

Published Jan 1, 2013• 5 citations
Publisher

Abstract

New fundamental (Galilean invariant) dynamical equations for four- component spinor wavefunctions are derived by using a different phase function than the Schrodinger phase function. The derived equations are called the generalized Levy-Leblond and generalized Schrodinger equa- tions because these equations reduce to the standard Levy-Leblond and Schrodinger equations when the Schrodinger phase function is used. The generalized and standard equations describe the same free elementary particles with spin 1/2, however, masses of the particles described by these equations are not the same as the generalized equations automati- cally account for particles with larger masses. This has important phys- ical implications as it shows that elementary particles with the same physical properties but different masses can result from using phase functions different than the Schrodinger phase function. Therefore, the main conclusion of this paper is that the existence of the three currently known families of elementary particles in the Standard Model of particle physics can be theoretically accounted for by choosing different phase functions.

DOI
10.12988/astp.2013.3672

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New fundamental (Galilean invariant) dynamical equations for four- component spinor wavefunctions are derived by using a different phase function than the Schrodinger phase function. The derived equations are called the generalized Levy-Leblond and generalized Schrodinger equa- tions because these equations reduce to the standard Levy-Leblond and Schrodinger equations when the Schrodinger phase function is used. The generalized and standard equations describe the same free elementary particles with spin 1/2, however, masses of the particles described by these equations are not the same as the generalized equations automati- cally account for particles with larger masses. This has important phys- ical implications as it shows that elementary particles with the same physical properties but different masses can result from using phase functions different than the Schrodinger phase function. Therefore, the main conclusion of this paper is that the existence of the three currently known families of elementary particles in the Standard Model of particle physics can be theoretically accounted for by choosing different phase functions.
A search for fundamental (Galilean invariant) dynamical equations for two and four-component spinor wave functions is conducted in Galilean space-time. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wave functions. A method to derive the Lévy-Leblond equation for a four-component spinor wave function is presented. It is formally proved that the Lévy-Leblond and Schrödinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrödinger phase factor. Physical implications of the obtained results and their relationships to the Pauli-Schrödinger equation are discussed.
A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Levy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Levy-Leblond and Schrodinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrodinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrodinger equation are discussed.
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.
The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.