Question

Repeat example 8.6 but assume that the upper state is the $2p_{\frac{1}{2}}$rather than the$\mathbf{2}{\mathit{p}}_{\frac{\mathbf{3}}{\mathbf{2}}}$

Short answer

The $1s_{\frac{1}{2}}$and $2p_{\frac{1}{2}}$states are both split into two levels.

Step-by-step solution

 Step 1: Concept of the ordering Rule for Z- projection.

The ordering rule for the quantum number of the z-projection of the total angular momentum $m_j$is:$m_j=-j,-j+1,....,j-1,j$

Here Jis the quantum number for the total angular momentum.

Determine the state of two levels which is split

The J 's for the$1s_{\frac{1}{2}}$ and$2p_{\frac{1}{2}}$states are both$\frac{1}{2}$since that's what the subscripts of the states are. Consequently, the's are given by:

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

The $1s_{\frac{1}{2}}$and$2p_{\frac{1}{2}}$states are both split into two levels.