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Annales de la fondation Louis de Broglie/Annales de la Fondation Louis de Broglie

Relativistic Pauli equation

Ayman H. Altorra

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

We derived relativistic Pauli equation depending on modified Dirac Hamiltonian and without using any kind of approximation. This equation affords relativistic entity to the electron magnetic moments, that enabled us to obtain straightforward formulation to relativistic Zeeman effect (ZE) which elucidates the very physical essence of the ZE.

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Journal de physique•Jan 1, 1977•7 citations
We derived relativistic Pauli equation depending on modified Dirac Hamiltonian and without using any kind of approximation. This equation affords relativistic entity to the electron magnetic moments, that enabled us to obtain straightforward formulation to relativistic Zeeman effect (ZE) which elucidates the very physical essence of the ZE.
Despite the success of the Dirac equation, the incorporation of the special theory of relativity into quantum mechanics predicts some paradoxes, like that the spin prediction from Dirac equation can be only identified with non- relativistic approximations (Pauli and Foldy-Wouthysen), as well as the spin predication is a relativistic quantum phenomena because the spin prediction is a necessary requirement of the relativistic quantum mechanics only. In this paper we show that the derivation of the spin and its magnetic moment can be done with a pure classical treatment. Since we start from the classical physical laws and the classical relativity principle to get the linear Schrödinger equation as a result the derivation of the spin and its magnetic moment can be done with a pure classical treatment. This approach result a Schrödinger equation, in which we show that the spin of the electron is a non relativistic quantum phenomenon too.
Relativistic extension of the Pauli Hamiltonian is ostensibly achieved by minimal coupling of electromagnetism to the free-particle Dirac Hamiltonian. But the free-particle Pauli Hamiltonian is pathology-free in its nonrelativistic domain, while the free-particle Dirac Hamiltonian yields completely fixed particle speed which is greater than c, spin orbit torque whose ratio to kinetic energy tends to infinity in the zero-momentum limit, and mega-violation of Newton's First Law in that limit. Furthermore, relativistic extension of the Pauli Hamiltonian is unique in principle because inertial frame hopping can keep the particle nonrelativistic. That extension is indeed readily achieved by upgrading the terms of the Pauli Hamiltonian's corresponding action to appropriate Lorentz invariants. The resulting relativistic Lagrangian yields a canonical momentum that can't be analytically inverted in general, but a physically-sensible successive-approximation scheme applies. For hydrogen and simpler systems approximation isn't needed, and the result, which includes spin-orbit coupling, is as transparently physically sensible as the relativistic Lorentz Hamiltonian is, a far cry from the Dirac Hamiltonian pathologies.
Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.
By viewing the electron as a wavepacket in the positive energy spectrum of the Dirac equation, we are able to achieve a much clearer understanding of its behavior under weak electromagnetic fields. The intrinsic spin magnetic moment is found to be established from the self-rotation of the wavepacket. A non-canonical structure is also exhibited in the equations of motion due to non- Abelian geometric phases of the Dirac spinors. The wavepacket energy can be expressed simply in terms of the kinetic, electrostatic, and Zeeman terms only. This can be transformed into an effective quantum Hamiltonian by a novel scheme, and reproduces the Pauli Hamiltonian with all-order relativistic corrections.
The increasing precision of experiments on the Zeeman effect of many electron atoms requires an ever increasing knowledge of relativistic and radiative effects. The generalization of the Breit equation as proposed by Hegstrom includes all these effects up to order c— 3. In the non relativistic limit the resulting Hamiltonian is quite intricate, but we show in this paper that it is possible to explain almost all of its terms within the frame of classical electrodynamics. In particular, picturing the atoms as point-like particles with intrinsic magnetic moments, we show that the physical phenomena underlying the Breit equation and its recent extension are mainly two well-known classical phenomena : i) the interaction of the magnetic moment of the constituent particles with the magnetic field they experience in their rest frame; ii) the Thomas precession of the spins of the accelerated particles.