Angular Momenta
An irreducible tensor $\tensor{T}^{(J)}_M$ obeys the following commutation relations with the angular momentum $\tensor{J}$:
\[\begin{equation} \begin{aligned} {}[J_{\pm},\tensor{T}^{(J)}_M] &= \sqrt{\frac{J(J+1)-M(M\pm1)}{2}}\tensor{T}^{(J)}_{M\pm1}, \\ [J_0,\tensor{T}^{(J)}_M] &= M\tensor{T}^{(J)}_M. \end{aligned} \label{eqn:ladder-operators} \tag{V3.1.1} \end{equation}\]
Orbital angular momentum
AngularMomentumAlgebra.OrbitalAngularMomentum — TypeOrbitalAngularMomentum()The angular momentum $\tensor{L}$ is associated with the spatial coordinates $\theta$ and $\phi$.
AngularMomentumAlgebra.rme — Methodrme(ℓ′, 𝐋̂, ℓ)Calculate the reduced matrix element of the orbital angular momentum:
\[\begin{equation} \tag{V13.2.67} \redmatrixel{\ell'}{\tensor{L}}{\ell} \defd \delta_{\ell'\ell}\sqrt{\ell(\ell+1)(2\ell+1)} \end{equation}\]
Examples
julia> rme(1, OrbitalAngularMomentum(), 1)
2.449489742783178
julia> rme(1, OrbitalAngularMomentum(), 2)
0AngularMomentumAlgebra.couplings — Methodcouplings(tensor::OrbitalAngularMomentum, ℓ)Generate all quantum numbers ℓ′ for which ⟨ℓ′||::OrbitalAngularMomentum||ℓ⟩ does not vanish.
Spin angular momentum
AngularMomentumAlgebra.SpinAngularMomentum — TypeSpinAngularMomentum()The spin angular momentum $\tensor{S}$ is the intrinsic angular momentum associated with the coordinate $s$.
AngularMomentumAlgebra.rme — Methodrme(s′, ::SpinAngularMomentum, s)Calculate the reduced matrix element of the spin angular momentum:
\[\begin{equation} \tag{V13.2.95} \redmatrixel{s'}{\tensor{S}}{s} \defd \delta_{ss'} \sqrt{s(s+1)(2s+1)} \end{equation}\]
Examples
julia> rme(half(1), SpinAngularMomentum(), half(1))
1.224744871391589
julia> rme(half(1), SpinAngularMomentum(), half(3))
0AngularMomentumAlgebra.couplings — Methodcouplings(tensor::SpinAngularMomentum, s)Generate all quantum numbers s′ for which ⟨s′||::SpinAngularMomentum||s⟩ does not vanish.
Total angular momentum
AngularMomentumAlgebra.TotalAngularMomentum — TypeTotalAngularMomentum()The total angular momentum $\tensor{J} = \tensor{L} + \tensor{S}$ results from the coupling of the orbital and spin angular momenta.
AngularMomentumAlgebra.rme — Methodrme((ℓ′,s′,J′), ::TotalAngularMomentum, (ℓ,s,J))Calculate the reduced matrix element of the total angular momentum:
\[\begin{equation} \tag{V13.2.38} \redmatrixel{\ell's'J'}{\tensor{J}}{\ell s J} \defd \delta_{\ell\ell'}\delta_{ss'}\delta_{JJ'} \sqrt{J(J+1)(2J+1)} \end{equation}\]
Examples
julia> rme((1,half(1),half(3)), TotalAngularMomentum(), (1,half(1),half(3)))
3.872983346207417
julia> rme((1,half(1),half(3)), TotalAngularMomentum(), (1,half(1),half(1)))
0AngularMomentumAlgebra.couplings — Methodcouplings(tensor::TotalAngularMomentum, (ℓ, s, J))Generate all quantum numbers ℓ′s′J′ for which ⟨ℓ′s′J′||::TotalAngularMomentum||ℓsJ⟩ does not vanish.