Motivated by the difficulties which persist in the construction of a consistent description of interacting relativistic particles with spin greater than 1, we examine the analogous problem within the framework of nonrelativistic quantum mechanics and find that all of the difficulties vanish in the (Galilei-invariant) limit. It is found that a unique, first-order, Galilei-covariant wave equation describing massive, spin-
s particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has
6s+1 components (
2s+1 of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin-
s particles so described are found to possess only an electric charge and a magnetic dipole with a
g factor of
s1. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-
s fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schr\"odinger theory is presented, as well as examples of theories which violate the minimality condition.
Shay and Good's wave equation is solved for a spin-1 particle with arbitrary magnetic dipole moment. Simultaneous eigenfunctions of the following three operators are used:
pz, the component of $\ensuremath{-}i\ensuremath{\nabla}$ in the direction of the field;
Jz, the component of x\ifmmode\times\else\texttimes\fi{}p+s; and
R02, the operator for the square of the distance to the center of the orbit in the projection of the motion perpendicular to the field. Explicit formulas for the allowed energies in terms of the quantum numbers are found. Determination of the wave functions is reduced to a set of linear algebraic equations.
establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.
It is shown that the Hilbert space of spin 12 particles, elementary under the Galilei group, decomposes into two subspaces invariant under the “static” Galilei group. This fact implies a linearization of the Schrödinger equation and suggests a square-root procedure to introduce electromagnetic interactions leading to the right expression for the intrinsic magnetic moment of the particle without using relativity. The procedure can be used to introduce the electron intrinsic magnetic moment in a more convincing way than is usual in textbooks.
In a purely nonrelativistic formulation, a Hamiltonian is obtained for the description of states of a charged particle with spin, interacting with an external electromagnetic field. For the specific cases of spin-1/2 and spin-1, the simplest choices for the Hamiltonian functions lead to a value for the (lowest order) intrinsic magnetic moment equal to the Bohr magneton in both cases. This is the same as is obtained in treatments that start from a relativistic formulation, as in the Dirac theory. The nonrelativistic and relativistic formulations both employ the ‘‘minimal coupling principle,’’ but the resulting Hamiltonians with couplings are, for very general considerations, not completely determined. However, the corresponding ambiguity that occurs in the nonrelativistic theory is also present in the relativistic formulation. The simpler nonrelativistic theory is more transparent in exhibiting these features of the problem and dispels the notion that spin (and the associated magnetic moment) require relativistic formulations for a proper understanding.