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

摘要

An investigation of the relativistic dynamics of N+1 spin-1/2 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 alpha~4mc~2.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 alpha~4mc~2 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_(nomel).The operator H_(nomel)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.