Confusion with some of the states of addition of angular momentum 1 +1/2 In the hydrogen atom, for $n=2$, I want to find all possible states.
For $l=0$ this is easy, and so it is with most of $l=1$. There are two states, however, that I don't understand.
They are:
$$\Psi_1 =|n=2;l=1;j=1/2;m_j=1/2>$$
$$\Psi_1 =|n=2;l=1;j=1/2;m_j=-1/2>$$
I find this incredibly confusing, because if $j=l+s$, then I get $1+1/2=1/2$, which doesn't make any sense. I know from the Clebsch-Gordon tables and from Griffith's book (Chapter intermediate Zeeman effect) that this state exists, but I can't wrap my head around it.
I would appreciate any explanation on how to understand this state
 A: 
if $j=l+s$, then I get $1+1/2=1/2$, which doesn't make any sense.

Yes, worded this way it doesn't make sense.
It seems you have misunderstood something.
When you have two angular momentum momenta (with $l$ and $s$),
then these can couple to give states with several different
total angular momenta ($j$):

*

*The largest possible value is $j_\text{max}=l+s$,

*the smallest possible value is $j_\text{min}=|l-s|$.

*and also the $j$ values in between (with step size $1$) are possible.

Summarizing this you have
$$j = l+s,\ l+s-1,\ l+s-2,\ ...,\ |l-s|.$$
For your example ($l=1$ and $s=\frac{1}{2}$) this boils down to
just two possible values of $j$:
$$j = \frac{3}{2},\ \frac{1}{2}.$$
You can visualize these two groups of states like this.
When $\vec{L}$ and $\vec{S}$ are nearly parallel then you get
the longest $\vec{J}$ ($j=\frac{3}{2}$). When they are nearly
antiparallel then you get the shortest $\vec{J}$ ($j=\frac{1}{2}$).
And for every $j$ there are $2j+1$ different states, from $m_j=+j$
(i.e. $\vec{J}$ pointing up) to $m_j=-j$ (i.e. $\vec{J}$ pointing down).

(image from Hyperphysics - Russell-Saunders or L-S coupling)
The $2$ states with $j=\frac{1}{2}$ are
(taken from Angular Momentum in the Hydrogen Atom
where this is worked out in detail):
$$\begin{align}
  \left|n=2;l=1;j=\frac{1}{2};m_j=+\frac{1}{2}\right> 
&=\sqrt{\frac{1}{3}}\left|n=2;l=1;m_l=0;m_s=+\frac{1}{2}\right> \\
&-\sqrt{\frac{2}{3}}\left|n=2;l=1;m_l=+1;m_s=-\frac{1}{2}\right> \\
 \left|n=2;l=1;j=\frac{1}{2};m_j=-\frac{1}{2}\right> 
&=\sqrt{\frac{2}{3}}\left|n=2;l=1;m_l=-1;m_s=+\frac{1}{2}\right> \\
&-\sqrt{\frac{1}{3}}\left|n=2;l=1;m_l=0;m_s=-\frac{1}{2}\right>
\end{align}$$
