Notation

Notation

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