What I know is total angular momentum $\vec{j}= \vec{\ell}+\vec{s}$, here $\ell=3$ for $f$ state, $s=1/2$ for spin, so the total angular momentum should be $7/2$, is this correct? is there any other allowed values?
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If $l=3$ and $s=1/2$ then the values of total angular momentum can take lie within the following range:
$$|l-s|\leq J\leq l+s$$
With $J$ incrementing by 1 between these two bounds.
$$\therefore 5/2\leq J\leq7/2$$
$J$ can take values $5/2$ and $7/2$
If you have a single electron system then your z component of total angular momentum can take on values -5/2,-3/2,-1/2, 1/2, 3/2, 5/2 for J=5/2 and -7/2,-5/2,-3/2,-1/2, 1/2, 3/2, 5/2, 7/2 for J=7/2. Overall you have 14 possible states your electron can be in.
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$\begingroup$ For more information on the inequality above, this is a good resource quantummechanics.ucsd.edu/ph130a/130_notes/node31.html $\endgroup$– AliCommented Nov 15, 2020 at 5:05