Relativistic dynamics of two spin-half particles in a homogeneous magnetic field

Abstract

Relativistic dynamics of two spin-1/2 particles in an external, homogeneous magnetic field is investigated here. The problem is important for a preliminary understanding of the effect of magnetic field on atoms and molecules at the relativistic level. The relativistic Hamiltonian is formulated in three distinct forms which involve the Bethe–Salpeter interaction, generalized Breit interaction and projected Breit interaction. The total pseudomomentum of the two-particle system is conserved in each case, and its components are distinct in the zero-charge sector. This permits the separation of the center of mass motion from the Hamiltonian of the neutral two-particle system. The resulting Hamiltonian operator describes the movement of the two particles in relative coordinates. It is further simplified by using suitable unitary transformations so as to reduce the one-particle operator for the first particle into a diagonal form. The effective equation of motion for the movement of the second particle in relative coordinates is then identified. A second set of transformations convert the two-particle relative Hamiltonian into a form where the one-particle operator for each spin-1/2 particle is completely diagonalized and separable into positive and negative energy states. The correspondingly transformed interaction operators can be written in an order by order expansion from which the odd terms are removable by using suitable Foldy–Wouthuysen type transformations in a systematic way. The resulting Hamiltonian operator reduces to previously known expressions when the magnetic field is switched off. Thus the two sets of transformations which convert the one particle parts completely into separable as well as diagonal forms also transform the interaction operator to generate terms consistently through order v2/c2. The field dependence lies entirely in the diagonalized one-particle parts, which is a consequence of the initial choice of interaction operators. Our results also include expressions corresponding to the interaction operator being projected. The Bethe–Salpeter and projected Breit cases lead to the same interaction operators for a hydrogen atom in the nonrelativistic limit. In the same limit the methodology directly yields the anomalous Zeeman interaction term, some correction to it, and terms which can account for nuclear magnetic resonance. All these terms are embedded in the final two-particle Hamiltonian operator. These, along with the previously known, field-independent, terms which describe the hyperfine interactions, can account for electronic and magnetic resonance spectroscopies on the basis of the same Hamiltonian.