Question

(a) Show that, taking into account the possible z-components of J, there are a total of 12 L S coupled states corresponding to 1 s 2 p in Table 8.3.

(b) Show that this is the same number of states available to two electrons occupying 1 s and 2 p if LS coupling were ignored.

Short answer

(a)There are 12 L Scoupled states when z component of J_tis included, up from the original 4.

(b) With the two possibilities from the 1 s electron, and the six possibilities from the 2 pelectron, that wills 12 total combinations.

Step-by-step solution

Explanation of total angular momentum quantum number

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.

$l_T$	St	Jt
1	0	1
1	1	0
1	1	1
1	1	2

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

Lt	St	Jt	MjT
1	0	1	-1,0,+1
1	1	0	0
1	1	1	-1,0,+1
1	1	2	-2, -1,0,+1,+2

Explanation of the total angular momentums of the two electrons.

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

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

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