Consider an orbital angular momentum l $($l is an integer and positive or zero$)$and a spin s$=\frac{1}{2}.$We now define a total angular momentum j $=l+s$ and we want to derive the states $|$j $=$ l $+1/2,$ mj $>$ with mj $$ l $+1/2.$ To do this, we consider the $$angular momentum descent operator$F_{-}=$ Fx $$ iFy for any angular momentum F$.$ We call F an angular momentum if it satisfies the commutator relations for angular momentum. Only from these commutator relations can we conclude for the eigenstates of such an angular momentum operator:

$F_{-}|$F$,$ mF $>=$ hbar $\sqrt[\phantom{2}]{F(F+1)-mF(mF-1)}|$F$,$ mF $1>$

We now want to apply this equation to the angular momenta $j,$ l and s$!$

$a7.$ Show that starting from $|$j $=$ l $+1/2,$ mj $=$ l $+1/2>=|$l$,$ ml $=$ l; s$,$ ms $=+1/2>$ by applying the operator j$=$ jx $$ ijy gives $(see$ picture$)$:

bHow can you analogously construct additional missing vectors of the subspace j $=$ l $+1/2?$