Exact solution of the relativistic dynamics of a spin-(1)/(2) particle moving in a homogeneous magnetic field

Abstract

The relativistic dynamics of one spin-1/2 particle moving in a uniform magnetic field is described by the Hamiltonian h(D)(0)(pi) = c alpha . pi + beta mc(2). The discrete (and semidiscrete) eigenvalues and the corresponding eigenspinors are in principle known from the work of Dirac, Rabi, and Bloch. These are extensively reviewed here. Next, exact solutions are worked out for the recoil dynamics in relative coordinates, which involves the Hamiltonian h(D)(0)(-k) = -c alpha . k + beta mc(2). Exact solutions are also explicitly calculated in the case where the spin-1/2 particle has an anomalous magnetic moment such that its Hamiltonian is given by h(D)(pi) = h(D)(0)(pi) - beta mu (ano)sigma . B. Similar exact solutions are derived here when the recoiling particle has an anomalous magnetic moment, that is, the eigenvalues and eigenspinors of the Hamiltonian h(D)(-k) = h(D)(0)(-k) - beta mu (ano)sigma . B are explicitly obtained. The diagonalized and separable form of the Hamiltonian hD(rr), written as (h) over tilde (D)(pi), has exceedingly simple forms of eigenspinors. Similarly, the diagonalized and separable form of the operator h(D)(-k), written as (h) over tilde (D)(-k), has very simple eigenspinors. The importance of these exact solutions is that the eigenspinors can be used as bases in a calculation involving many spin-1/2 particles placed in a uniform magnetic field. (C) 2001 , Inc.