Relativistic dynamics of half-spin particles in a homogeneous magnetic field: An atom with nucleus of spin 1∕2

Abstract

An investigation of the relativistic dynamics of N+1 spin-12 particles placed in an external, homogeneous magnetic field is carried out. The system can represent an atom with a fermion nucleus and N electrons. Quantum electrodynamical interactions, namely, projected Briet and magnetic interactions, are chosen to formulate the relativistic Hamiltonian. The quasi-free-particle picture is retained here. The total pseudomomentum is conserved, and its components are distinct when the total charge is zero. Therefore, the center-of-mass motion can be separated from the Hamiltonian for a neutral (N+1)-fermion system, leaving behind a unitarily transformed, effective Hamiltonian H(0) at zero total pseudomomentum. The latter operator represents the complete relativistic dynamics in relative coordinates while interaction is chosen through order alpha4mc2. Each one-particle part in the effective Hamiltonian can be brought to a separable form for positive- and negative-energy states by replacing the odd operator in it through two successive unitary transformations, one due to Tsai [Phys. Rev. D 7, 1945 (1973)] and the other due to Weaver [J. Math. Phys. 18, 306 (1977)]. Consequently, the projector changes and the interaction that involves the concerned particle also becomes free from the corresponding odd operators. When this maneuver is applied only to the nucleus, and the non-Hermitian part of the transformed interaction is removed by another unitary transformation, a familiar form of the atomic relativistic Hamiltonian H(atom) emerges. This operator is equivalent to H(0). A good Hamiltonian for relativistic quantum chemical calculations, H(Qchem), is obtained by expanding the nuclear part of the atomic Hamiltonian through order alpha4mc2 for positive-energy states. The operator H(Qchem) is obviously an approximation to H(atom). When the same technique is used for all particles, and subsequently the non-Hermitian terms are removed by suitable unitary transformations, one obtains a Hamiltonian H(T) that is equivalent to H(atom) but is in a completely separable form. As the semidiscrete eigenvalues and eigenfunctions of the one-particle parts are known, the completely separable Hamiltonian can be used in computation. A little more effort leads to the derivation of the correct atomic Hamiltonian in the nonrelativistic limit, H(nonrel). The operator H(nonrel) is an approximation to H(T). It not only retains the relativistic and radiative effects, but also directly exhibits the phenomena of electron paramagnetic resonance and nuclear magnetic resonance.