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, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$ , 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 $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ 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 $Δm Δϕ$ , $Δj Δχ$ , 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 $j_m (ϕ,θ,χ)$ 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, $$. Interestingly, for low j, even down to $j=1/2$ , these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$, $θ$, $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$, $Δj Δχ$, and $ΔϕΔθ$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $$. 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 $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.

arXiv:2305.11456May 19, 20230 citations

Unified treatment of the total angular momentum of single photons via generalized quantum observables

Mário Motta, Giacomo Guarnieri, L. Lanz +1 more

In this paper, we provide a consistent framework to address the notorious difficult decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component.We discuss the canonical decomposition into SAM and OAM components, which are the generators of internal and spatial rotations in the space of physical states.We find that those operators are mutually compatible but unsharp quantum observables, therefore Positive Operator-Valued Measures describe their joint measurements.We present another decomposition of the TAM, which we denote as a non-canonical one.The operators resulting from this decomposition are mutually incompatible but sharp quantum observables, thus Projector-Valued Measurements.This fact reflects their consistency with the transversality condition of single-photon wavefunctions, thus explains the underlying physics from a quantum information theoretic view.Furthermore, we discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.The difference between the canonical and non-canonical momenta leads to observable differences in higher-order statistical moments.In this paper, we provide a consistent framework to address the notorious difficult decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component.We discuss the canonical decomposition into SAM and OAM components, which are the generators of internal and spatial rotations in the space of physical states.We find that those operators are mutually compatible but unsharp quantum observables, therefore Positive Operator-Valued Measures describe their joint measurements.We present another decomposition of the TAM, which we denote as a non-canonical one.The operators resulting from this decomposition are mutually incompatible but sharp quantum observables, thus Projector-Valued Measurements.This fact reflects their consistency with the transversality condition of single-photon wavefunctions, thus explains the underlying physics from a quantum information theoretic view.Furthermore, we discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.The difference between the canonical and non-canonical momenta leads to observable differences in higher-order statistical moments.

arXiv:1806.05479New Journal of PhysicsDec 12, 20182 citations

Angular Momentum Operators and Their Matrix Elements

Springer eBooksJan 1, 20020 citations

Factorizations With Application To The Momentum Representation

Owen L. de Lange, R. E. Raab

Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.

Nov 28, 19910 citations

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,

j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(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

cos θ_m=(m/|j|)

is the Vector-Model polar angle. The

j_m (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

Δm Δϕ

Δj Δχ

, 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

j_m (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

j_m (ϕ,θ,χ)

gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.