.

We have been writing |x> where x was ``hi'' or ``lo''. But there are two quantum numbers involved here, one associated with L²and the other associated with L_z, so we need to expand the ket to include two slots. By convention, the first slot is reserved for L² and the second for L_z. Thus we have, so far, $\vert\ell,?>$$">.

Now we need to work on laddering down from the hi ket or up from the low one, looking at L_z. We have

$$L_z\vert\ell, quantum\ number> = something\vert\ell,quantum\ number>$$ = something\vert\ell,quantum\ number> \end{displaymath}">

specifically,

$$L_z\vert\ell, m_\ell> = m_\ell \hbar\vert\ell,m_\ell>$$ = m_\ell \hbar\vert\ell,m_\ell> \end{displaymath}">

and then, from $L^-L_z\vert\ell,hi> = K' \hbar\vert\ell,hi>$ = K' \hbar\vert\ell,hi>$">:

$$L^- L_z\vert\ell, m_\ell> = m_K \hbar L_-\vert\ell,m_\ell>$$ = m_K \hbar L_-\vert\ell,m_\ell> \end{displaymath}">