If we apply the rising/lowering operators for angular momentum to a state $|{l,m}\rangle$, we get:

\begin{align} L_+|{l,m}\rangle = C_+(l,m)|{l,m+1}\rangle \\ L_-|{l,m}\rangle = C_-(l,m)|{l,m-1}\rangle \end{align}

it can be shown that: \begin{align} C_+(l,m) = \hbar\sqrt{(l-m)(l+m+1)}\\ C_-(l,m) = \hbar\sqrt{(l+m)(l-m+1)} \end{align}


\begin{align} L_+|{l,m}\rangle = \hbar\sqrt{(l-m)(l+m+1)}|{l,m+1}\rangle \\ L_-|{l,m}\rangle = \hbar\sqrt{(l+m)(l-m+1)}|{l,m-1}\rangle \end{align}

On the other hand:

\begin{align} \langle{L_\pm(l,m)|L_\pm(l,m)}\rangle \geq 0 \end{align}

And it follows from (7) that:

\begin{align} \langle{L_\pm(l,m)|L_\pm(l,m)}\rangle &= \langle{l,m|L_\mp L_\pm|l,m}\rangle\\ & = \langle{l,m|L^{2} - L_{z}^{2} \pm\hbar L_z|l,m}\rangle\\ & = \hbar^2 [l(l+1) - m(m\mp 1)] \geq 0 \end{align}

This implies that both:

\begin{align} l(l+1) - m(m- 1) \geq 0\\ l(l+1)- m(m+ 1) \geq 0 \end{align}

Now here comes my doubt, gasiorowicz's book (Quantum Physics, Thrid Edition, Chapter 7) claims: Since $l\geq 0$ It follows form the above that:

\begin{align} -l\leq m \leq l \end{align}

But i cant see how the above inequalities imples the last one. Any help would be appriciate, thanks.


If $0\leq l<m$, then $l+1<m+1$, so multiplying these inequalities yields \begin{align} l(l+1)<m(m+1) \end{align} (we can do this since everything is non-negative), so this violates the second inequality you wrote down.

Next, suppose $m<-l$, which means $0\leq l<-m$, and thus by the same reasoning as above, we have \begin{align} l(l+1)&< (-m)[(-m)+1]\\ &=m(m-1) \end{align}

which violates your first inequality.

Therefore, we must have $-l\leq m\leq l$.

  • $\begingroup$ Thank you!, this was very helpful, now i understand where the last inequality came from. $\endgroup$ Sep 4 '21 at 21:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.