What is the motivation for the values of the magnetic quantum number $m_l$ to take values of $ -l, -l+1, \cdots , l $ where $l$ is the angular momentum number? The ladder operators for angular momentum do imply that is some value $a\hbar$ is an eigenvalue of $L_z$ then so too is $(a\pm 1)\hbar$ provided the limiting values are not reached. Some constraint for these limiting values can be found from considering that $\langle L^2\rangle \geq \langle L_z\rangle$; that is, the magnitude of $m_l$ is bounded by $\sqrt{l(l+1)}$. However none of these considerations eliminate the possibility of $L_z$ having intermediate eigenvalues as well as the integer incremental steps suggested from the ladder operators. They also do not motivate why $l$ is integral and why the maximum values of $m_l$ ate given by $l$ and not, say, $\sqrt{l(l+1)}$ exactly.


This happens because of the commutator between $L_z$ and $L_{\pm}$.

$$[L_z,L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$

The above can be easily shown from the angular momentum algebra $[L_i,L_j]=\epsilon_{ijk}L_jL_k$ and from the definition of $L_{\pm}$ operators (that is, $L_{\pm}=L_x\pm iL_y$).

Here's how to conclude about the $\hbar$ separated spectrum of $L_z$:

Let $|l,m_{l}\rangle$ be an eigenket of $L_z$ with eigenvalue $m_l\hbar$. One can use (1) to show that , $L_{\pm}|l,m_{l}\rangle$ is an eigenket of $L_z$ as well.

$$L_z(L_{\pm}|l,m_l\rangle)=\underbrace{(L_zL_{\pm}-L_{\pm}L_z)}_{\pm \hbar L_{\pm}}|l,m_l\rangle +L_{\pm}\underbrace{L_z|l,m_l\rangle}_{m_l|l,m_l\rangle}=(m_l\pm \hbar)L_{\pm}|l,m_l\rangle$$

Thus, we find that $L_{\pm}$ creates a ladder of eigenstates for $L_{z}$ where the "rung" separation is exactly $\hbar$. One can further show that there exists a bottom rung and a top rung and thus they get bounded.


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