My aim is to prove the restriction for the orbital momentum quantum number $-\ell \leq m \leq \ell$. My professor was giving me the hint, that I should use the norm of the state $|| L_+ |\ell,m\rangle ||$ with the state vector $|\ell, m\rangle$.
So, I start with the norm
$$|| L_+ |\ell,m\rangle ||^2 = \langle\ell,m| L_-L_+ |\ell,m\rangle \geq 0.$$
First of all, I calculate the operator product:
$$L_-L_+ = (L_x - iL_y)(L_x + i L_y) = L_x^2 + L_y^2 + i[L_x, L_y] = \vec{L}^2 - L_z^2 - \hbar L_z.$$
Using the eigenvalues of the operator, I get
$$|| L_+ |\ell,m\rangle || ^2 = \underbrace{\langle\ell, m|\ell,m\rangle}_{=1} \, \hbar^2 \big( \ell(\ell+1) - m(m+1) \big) \geq 0.$$
Finally I have
$$\ell(\ell + 1) \geq m(m+1).$$
Calculating now the norm $|| L_- |\ell, m\rangle ||$ leads me to a similar inequality:
$$\ell(\ell + 1) \geq m(m-1).$$
My question now is, how do get from these two inequalities to the restriction $-\ell \leq m \leq \ell$?
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