Consider an orbital angular momentum l $($l is an integer and positive or zero$)$ and a spin s with s $=1/2.$ The tensor product $|$l$,$ ml ; s$,$ ms$>=|$l$,$ ml$>|$s$,$ ms$>$ is then a basis of the state space, which is formed from common eigenstates of $l^2,l_z,s^2$ and $s_z.$ We now define a total angular momentum j $=l+$ s and construct a new basis of the state space, which this time consists of simultaneous eigenstates of $j^2$ and $j_z.$ Now there must be a basis of common eigenfunctions of the four operators $j^2,j_z,s^2,l^2.$ In the following we will call this the $$total angular momentum basis$.$ We now want to construct this new basis from the tensor product basis.

a First show: $jz|$l$,$ ml; s$,$ ms$>=$ hbar$($ml $+$ ms$)|$l$,$ ml; s$,$ ms$>!$

$b$ Now justify that $|$l$,$ ml $=$ l; s$,$ ms $=+1/2>$ from the tensor product basis is the basis vector $|$j $=$ l $+1/2,$ mj $=$ l $+1/2>$of the total angular momentum basis!