Angular-momentum basis and tensor labels

PDF p. 3: Atomic and molecular systems, opening paragraph and equation (10)

A spin label is $J\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$, and its magnetic labels are $m=-J,-J+1,\dots ,J$. In the finite spin-$J$ block $V_J={\mathbb{C}}^{2J+1}$, write the specified orthonormal basis vector $e_{J,m}$ as $\mathrm{\mid }J,m⟩$. The operators $J_x,J_y,J_z$ are the block operators, and ${\mathbf{J}}^2=J_x^2+J_y^2+J_z^2$. Thus ${\mathbf{J}}^2\mathrm{\mid }J,m⟩=J(J+1)\mathrm{\mid }J,m⟩$ and $J_z\mathrm{\mid }J,m⟩=m\mathrm{\mid }J,m⟩$; in particular $\mathrm{\mid }m\mathrm{\mid }\le J$.

For $J\ge 1$ and $\delta \in \{-1,0,1\}$, the article's label $C_{1,\delta ;J,m}^{J-1,m+\delta }$ denotes the Clebsch–Gordan coefficient in the ordered coupling $V_1\otimes V_J$: it is the coefficient of $\mathrm{\mid }1,\delta ⟩\otimes \mathrm{\mid }J,m⟩$ in the normalized coupled vector of total spin $J-1$ and magnetic label $m+\delta$. A coupled vector is obtained from a chosen unit highest-weight vector in that spin summand by the supplier's positively normalized lowering convention. If $m+\delta$ is outside the magnetic labels of spin $J-1$, this coefficient is taken to be zero. The order of factors and the chosen highest-weight phase are part of the convention; comparison with a numerical coefficient table requires matching both. The article uses these labels in its rank-one transition operators.

Finite operator and ket conventions

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

The finite spin-block construction supplies the orthonormal basis and operators and gives the two displayed eigenvalue identities. The finite operator conventions give the meaning of the ket and ordered tensor-product notation. For $J\ge 1$, the spin-composition construction includes the $J-1$ summand in $V_1\otimes V_J$ and supplies its normalized coupled vectors for admissible magnetic labels. Their unique coefficients in the product basis are therefore well defined after the highest-weight vector is chosen. Extending the coefficient by zero outside that label range is an explicit convention. Changing the highest-weight phase multiplies all coefficients in the summand by the same phase, so the stated phase qualification is necessary. Finite operator and ket conventions Finite spin blocks, spherical tensors and Clebsch–Gordan phases