I have a doubt regarding the calculation of total angular momentum of electron in an atom. Which is the right way to do it?
Method 1:
Total magnetic moment
$$
\begin{align}
\vec{\mu_J} &= \vec{\mu_L}+\vec{\mu_S}
\\&= g_L \mu_B\vec{L}+g_s \mu_B\vec{S}.
\end{align}
$$
Since $g_L = -1$ and $g_S=-2$,
$$\begin{align}\vec{\mu_J}& = -\mu_B\vec{L}-2\mu_B\vec{S}
\\&= -\mu_B(\vec{L}+2\vec{S}),\end{align}$$
where
$$|\mu_J|=\mu_B|\vec{L}+2\vec{S}|$$
and
$$|\mu_J|=\mu_B\sqrt{|\vec{L}|^2+4|\vec{S}|^2+4\vec{L}.\vec{S}}.$$
Method 2:
Here we calculate Landé $g$ factor as
$$g_J=1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)},$$
and then substitute in the equation:
$$|\mu_J| = g \frac{e\hbar}{2m}\sqrt{j(j+1)}.$$
I wanted to know what is wrong with method 1.