Abstract We present an alternative formulation of quantum mechanical angular momentum, based on spatial wavefunctions that depend on the Euler angles φ , θ , χ , and have an additional internal projection n . The wavefunctions are Wigner D-functions, D nm s ( φ , θ , χ ) , for which the body-fixed projection quantum number n has the unusual value n = s = s s + 1 = s , or n = 0 . We show that the states D s m s ( φ , θ , χ ) of elementary particles with spin s give a gyromagnetic ratio of g = 2 for s > 0 , and we identify these as the spatial angular-momentum wavefunctions of known fundamental charged particles with spin. All known Standard-Model particles can be categorized with either value n = s or n = 0 , and all known particle reactions are consistent with the conservation of its projection in the internal frame, and with internal-frame Clebsch-Gordan coefficients of unity. Therefore, we make the case that the D nm s ( φ , θ , χ ) are useful as spatial wavefunctions for spin. Some implications and new predictions related to the quantum number n for fundamental particles are discussed, such as the proposed Dirac-fermion nature of the neutrino, the explanation of some Standard-Model structure, and some proposed dark-matter candidates.
In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum
j. In contrast, here we show that a spatial wavefunction,
jm(ϕ,θ,χ)= eimϕδ(θ−θm) ei(j+1/2)χ, which treats
j in the
∣jm> state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators;
ϕ,
θ,
χ are the Euler angles, and
cosθm=(m/∣j∣) is the Vector-Model polar angle. The
jm(ϕ,θ,χ) gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in
j and
m) which predicts the effective wavepacket angular uncertainty relations for
ΔmΔϕ,
ΔjΔχ, and
ΔϕΔθ, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive
j-rotation measurements. We also use the
jm(ϕ,θ,χ) to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles,
g=2, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to
j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that
jm(ϕ,θ,χ) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.
Abstract The concept of angular momentum in quantum mechanics has many counterintuitive aspects, such as assuming only certain discrete values and only certain orientations in space in the presence of external fields. Spin is more esoteric yet; it is hard to imagine how point‐particles with no spatial extent can have an intrinsic angular momentum. Nevertheless, spin is the most fundamental concept in magnetic resonance. Appreciation of the spin concept requires some understanding of the relativistic quantum mechanics developed by P. A. M. Dirac. The history of the evolution of the spin concept is traced—from the turn‐of‐the‐century observation of the Zeeman effect, through the old and new quantum mechanics, and on to Dirac's theory of the electron, which was published in the late 1920s. The special relativity theory is discussed, and there is a brief explanation of how spin emerges naturally in a relativistically correct formulation of quantum mechanics.
The chapter begins with the introduction of the two-particle Schrödinger wave equation (SWE) and the solution of this equation for the hydrogen atom. The orbital angular momentum of the electron results from the SWE solution. The Pauli spinors are introduced, and the SWE wavefunctions are modified to account for the spin of the electron. The structure of multielectron atoms is then discussed. The discussion is focused on low-Z atoms for which Russell–Saunders or LS coupling is appropriate. Alternate coupling schemes are briefly discussed. Angular momentum coupling algebra, the Clebsch–Gordan coefficients, and 3j symbols are then introduced. The Wigner–Eckart theorem is discussed, and the use of irreducible spherical tensors for evaluation of quantum mechanical matrix elements is discussed in detail.
One of the most important applications of the quantum mechanical formalism developed in the previous chapter is to angular momentum. This is because many microsystems are spherically symmetric, or approximately so, and angular momentum is therefore conserved. This chapter discusses that a careful study of quantized angular momentum leads to a completely new physical phenomenon – intrinsic spin. The fact that the angular momentum vector of a quantum particle must point in certain restricted directions is a bizarre quantum property. In some respects the abstract idea of an internal angular momentum is reminiscent of the classical concept of a spinning body. In classical mechanics angular momentum is added vectorially but in the quantum case the issue is more subtle. The procedure will be illustrated by the example of the addition of two spins. As in classical mechanics a charged quantum system with angular momentum possesses a magnetic moment μ.
The adaptation of Wigner's induced representation for a relativistic quantum theory making possible the construction of wave packets and admitting covariant expectation values for the coordinate operator x μ introduces a foliation on the Hilbert space of states. The spin-statistics relation for fermions and bosons implies the universality of the parametrization of orbits of the induced representation, implying that all particles within identical particle sets transform under the same SU (2) subgroup of the Lorentz group, and therefore their spins and angular momentum states can be computed using the usual Clebsch–Gordan coefficients associated with angular momentum. Important consequences, such as entanglement for subsystems at unequal times, covariant statistical correlations in many-body systems and the construction of relativistic boson and fermion statistical ensembles, as well as implications for the foliation of the Fock space and for quantum field theory are briefly discussed.