Question

For hydrogen, an electron is in 2p state. Obtain the magnitude of orbital angular momentum and possible

z-components of orbital angular momentum 𝐿⃗ . What would be the possible values of total angular

momentums (𝐽 ) and associated magnetic moments 𝜇𝐽

⃗⃗⃗

Answer 1

For P

State

l=1,s=$\frac{1}{2}$

Calculate J value

L-s to L+s

Total anguler momentum

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

Z- components of anguler momentum

m_l= $\frac{-3}{2},\frac{-1}{2},\frac{1}{2},\frac{3}{2}$

$\\J=L-\frac{1}{2}$

We know that formula total angular moment

$\mu=g_{lp}(j-\frac{1}{2})+\frac{g_{sp}}{2}$

$g_{lp}=1,g_{sp}=5.585\mu_N$

$\mu=1\times(\frac{3}{2}-\frac{1}{2})+\frac{5.585}{2}$

total angular

momentum (𝐽 ) and associated magnetic moment

$\mu=1+2.7925=3.7925\mu_N$

$J=L-\frac{1}{2}$

$\mu=\frac{j}{j+1}(g_{lp}(j+\frac{3}{2})-\frac{g_{sp}}{2})$

$\mu=\frac{\frac{1}{2}}{\frac{1}{2}+1}(1\times(\frac{1}{2}+\frac{3}{2})-\frac{5.585}{2})$

$$\mu=\frac{1}{3}(2-\frac{5.585}{2})=-0.264\mu_N$$