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.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
Quantum-mechanical spin is often thought of in terms of classical angular momentum. In fact spin is defined by its commutation relations and the spin and orbital angular momentum operators are very different. Here we solve the Dirac equation in a rotating frame of reference and create a localized wave packet from the resulting wave functions to examine the consequences of the rotational motion. We highlight some unexpected effects in the properties of the wave packet and show that the spin operator and orbital angular momentum operator describe different aspects of the rotational properties of the wave packet. It is also observed that neither quantum-mechanical spin nor orbital angular momentum can be fully understood within an inertial frame.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the later cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrodinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.
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.