91Ó°ÊÓ

We need ordering rules for $J_T$and $m_{jT}$are needed.

The ordering rules for ${\mathit{j}}_{\mathbf{T}}$are.

$\mathit{j}\mathit{T}\mathbf{=}\left|l_T-S_T\right|\mathbf{,}\left|l_T-s_T\right|\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{l}}_{\mathbf{T}}\mathbf{+}{\mathit{s}}_{\mathbf{T}}\mathbf{-}\mathbf{1}\mathbf{,}{\mathit{l}}_{\mathbf{T}}\mathbf{+}{\mathit{s}}_{\mathbf{T}}$

Where $l_T$represents quantum number of total orbital angular momentum and $s_T$represents quantum number of total spin angular momentum.

The ordering rules for $m_{jT}$are:

$m_{jT}=-J_T,-J_T+1,....,J_T-1,J_T$

Where $J_T$represents total angular momentum quantum number

Calculation:

The available LS couplings for the 1 s 2 p state are given as.

But that can be expanded to show the M_jT possibilities as well, given the ordering rules for that.

$m_j=-\frac{1}{2},+\frac{1}{2}$

The electron in the 2 p shell will have an I of 1 , meaning that can have the values.

$j=\frac{1}{2},\frac{3}{2}$

Number of possibilities can be shown with ignoring L S coupling as well as checkingthe combinations available by considering the total angular momentums of the twoelectrons

Therefore, its values for $m_j$will be

$$\begin{gathered} m_j\left(j=\frac{1}{2}\right)=-\frac{1}{2}+\frac{1}{2} \\ {m}_j\left(j=\frac{3}{2}\right)=-\frac{3}{2},-\frac{1}{2},+\frac{1}{2},+\frac{3}{2} \end{gathered}$$