# Notation

## Notation

• $\mathbf {L}$ denotes the total orbital angular momentum $\mathbf {L} =\sum _{i}\mathbf {l_{i}}$ , with $\mathbf {l_{i}}$ single-particle angular momentum. The eigenvalues of $L^{2}$ are such that $L^{2}|\varphi >=\hbar ^{2}L(L+1)|\varphi >$ and are simply denoted by the letter L. The eigenvalues of $\mathbf {L_{z}}$ are such that $\mathbf {L_{z}} |\varphi >=\hbar m_{L}|\varphi >$ and are simply denoted by the letter $m_{L}$ • $\mathbf {S}$ denotes the total spin $\mathbf {S} =\sum _{i}\mathbf {s_{i}}$ , with $\mathbf {s_{i}}$ single-particle spin.The eigenvalues of $S^{2}$ are such that $S^{2}|\varphi >=\hbar ^{2}S(S+1)|\varphi >$ and are simply denoted by the letter S. The eigenvalues of $\mathbf {S_{z}}$ are such that $\mathbf {S_{z}} |\varphi >=\hbar m_{S}|\varphi >$ and are simply denoted by the letter $m_{S}$ • $\mathbf {J}$ denotes the total angular momentum $\mathbf {j_{i}} =\mathbf {l_{i}} +\mathbf {s_{i}}$ , $\mathbf {J} =\sum _{i}\mathbf {j_{i}}$ .The eigenvalues of $J^{2}$ are such that $J^{2}|\varphi >=\hbar ^{2}J(J+1)|\varphi >$ and are simply denoted by the letter J. The eigenvalues of $\mathbf {J_{z}}$ are such that $\mathbf {J_{z}} |\varphi >=\hbar m_{J}|\varphi >$ and are simply denoted by the letter $m_{J}$ • Keep in mind the addition rules for angular momenta: let us call $\mathbf {J_{1}} ,\mathbf {J_{2}}$ two arbitrary angular momenta.
{\begin{aligned}\mathbf {J_{tot}} =\mathbf {J_{1}} +\mathbf {J_{2}} \\m_{J_{tot}}=m_{J_{1}}+m_{J_{2}}\\|J_{1}-J_{2}|\leq J_{tot}\leq J_{1}+J_{2}\\-J_{tot}\leq m_{J_{tot}}\leq +J_{tot}\end{aligned}} 