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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Nonrelativistic Quantum Mechanics for Particles with Arbitrary Spin

William J. Hurley

Published May 15, 1971• 37 citations
Publisher

Abstract

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 $\frac{1}{s}$. 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.

DOI
10.1103/physrevd.3.2339

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IntechOpen eBooks•Sep 30, 2020•8 citations
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-sss particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has 6s+16s+16s+1 components (2s+12s+12s+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-sss particles so described are found to possess only an electric charge and a magnetic dipole with a ggg factor of 1s\frac{1}{s}s1​. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-sss 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.
The magnetic moment of a particle of nonzero spin SSS is computed by the introduction of minimal coupling into the appropriate Galilean-invariant wave equation. It is shown that the requirement that the differential equations be of first order, together with a minimality assumption on the number of components, uniquely implies a ggg factor of 1S\frac{1}{S}S1​. The role played by this minimality condition is made explicit by means of a counterexample.
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.
It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in ${L}^{2}(\ensuremath{\Gamma})$ space. In a sharp-point limit $s\ensuremath{\rightarrow}+0$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space Mm{M}_{m}Mm​, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\ensuremath{\rightarrow}+0$ of the relativistic SSS matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.
Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.