.

(*restored equation*)

$$(L^-L^+ + \hbar L_z + L_z^2)\vert>= L^- L^+\vert> + \hbar L_z\vert> + L_z^2\vert>= K' \hbar^2 \vert>$$= L^- L^+\vert> + \hbar L_z\vert> + L_z^2\vert>= K' \hbar^2 \vert> \end{displaymath}">

If this particular ket is the highest one, |hi>, so that

$$L_z\vert hi> = K_{hi} \hbar\vert hi>$$ = K_{hi} \hbar\vert hi> \end{displaymath}">

then laddering up on it must result in destruction, so we have if

$$L^- L^+\vert hi> \rightarrow 0$$ \rightarrow 0 \end{displaymath}">

$$+ \hbar L_z\vert hi> + L_z^2\vert hi>= K' \hbar^2 \vert hi>$$ + L_z^2\vert hi>= K' \hbar^2 \vert hi> \end{displaymath}">

$$+ (K_{hi}) \hbar^2\vert hi> + (K_{hi})^2 \hbar^2\vert hi>= K' \hbar^2 \vert hi>$$ + (K_{hi})^2 \hbar^2\vert hi>= K' \hbar^2 \vert hi> \end{displaymath}">

which means that

+ (K_hi) + (K_hi)² = K'

while, working down from the low ket (using Equation 44, one has from

$$L^2\vert lo> = K' \hbar^2\vert lo>$$ = K' \hbar^2\vert lo> \end{displaymath}">

by substitution,

$$L^+ L^-\vert lo> - \hbar L_z\vert lo> + L_z^2\vert lo>= K' \hbar^2 \vert lo>$$ - \hbar L_z\vert lo> + L_z^2\vert lo>= K' \hbar^2 \vert lo> \end{displaymath}">

which is

$$- (K_{lo}) \hbar ^2L_z\vert lo> + (K_{lo})^2 \hbar^2\vert lo>= K' \hbar^2 \vert lo>$$ + (K_{lo})^2 \hbar^2\vert lo>= K' \hbar^2 \vert lo> \end{displaymath}">

since

$$L^+ L^-\vert lo> \rightarrow 0$$ \rightarrow 0 \end{displaymath}">

i.e.,

- (K_lo) + (K_lo)² = K'

which means

- (K_lo) + (K_lo)² = K' = + (K_hi) + (K_hi)² = (K_lo)((K_lo) - 1) = (K_hi)((K_hi)+1)

which can only be true if ( K_lo) = - (K_hi). Say (K_lo) = 7, then K' = 7*8 and -7(-7-1) which is the same! Note that if (K_hi) = 7.1 then (K_lo) can not be achieved by stepping down integral multiples of $\hbar$. Thus if (Khi) were 2.1, then 2.1 $\rightarrow$ 1.1 $\rightarrow$ 0.1 $\rightarrow$-0.9 $\rightarrow$ -1.9 $\rightarrow$ -2.9 which misses -2.1!

We conclude from Equation 46 that if (K_hi) = $\ell$, then

$$K' = K(\ell+1)$$