.

(*restored equation*)

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

which, upon using the appropriate commutator gives

$$(L_zL^-+\hbar) \vert\ell, m_\ell> = m_\ell \hbar L_-\vert\ell,m_\ell>$$ = m_\ell \hbar L_-\vert\ell,m_\ell> \end{displaymath}">

or

$$(L_zL^-) \vert\ell, m_\ell> = (m_\ell \hbar -\hbar) L_-\vert\ell,m_\ell>$$ = (m_\ell \hbar -\hbar) L_-\vert\ell,m_\ell> \end{displaymath}">

which means that laddering down on an eigenket of L_z results in another eigenket of L_z whose eigenvalue is lower than the original one by $\hbar$.