Angular Momentum Algebra: Raising and Lowering Operators

We have already derived the commutators of the angular momentum operators

$$\begin{eqnarray*}[L_x,L_y]&=&i\hbar L_z \\ {[L_i,L_j]}&=&i\hbar\epsilon_{ijk}L_k \\ {[L^2,L_i]}&=&0 .\\ \end{eqnarray*}$$

We have shown that angular momentum is quantized for a rotor with a single angular variable. To progress toward the possible quantization of angular momentum variables in 3D, we define the operator $L_+$ and its Hermitian conjugate $L_-$.

$$\bgroup\color{black} L_\pm\equiv L_x\pm iL_y .\egroup$$

Since $L^2$ commutes with $L_x$ and $L_y$, it commutes with these operators.

$$\bgroup\color{black} [L^2,L_\pm]=0 \egroup$$

The commutator with $L_z$ is.

$$\bgroup\color{black} [L_\pm,L_z]=[L_x,L_z]\pm i[L_y,L_z]=i\hbar(-L_y\pm iL_x)=\mp\hbar L_\pm .\egroup$$

From the commutators $[L^2,L_\pm]=0$ and $[L_\pm,L_z]=\mp\hbar L_\pm$, we can derive the effect of the operators $L_\pm$ on the eigenstates $Y_{\ell m}$, and in so doing, show that $\ell$ is an integer greater than or equal to 0, and that $m$ is also an integer

$$\ell=0,1,2,...$$

$$-\ell\leq m\leq\ell$$

$$m=-\ell, -\ell+1,...,\ell$$

$$L_\pm Y_{\ell m}=\hbar\sqrt{\ell(\ell+1)-m(m\pm 1)}Y_{\ell(m\pm 1)}$$

Therefore, $L_+$ raises the $z$ component of angular momentum by one unit of $\hbar$ and $L_-$ lowers it by one unit. The raising stops when $m=\ell$ and the operation gives zero, $L_+Y_{\ell \ell}=0$. Similarly, the lowering stops because $L_-Y_{\ell -\ell}=0$.

Angular momentum is quantized. Any measurement of a component of angular momentum will give some integer times $\hbar$. Any measurement of the total angular momentum gives the somewhat curious result

$$\bgroup\color{black} \vert L\vert=\sqrt{\ell(\ell+1)}\hbar \egroup$$

where $\ell$ is an integer.

Note that we can easily write the components of angular momentum in terms of the raising and lowering operators.

$$\begin{eqnarray*} L_x&=&{1\over 2}(L_+ +L_-) \\ L_y&=&{1\over 2i}(L_+ -L_-) \\ \end{eqnarray*}$$

We will also find the following equations useful (and easy to compute).

$$\begin{eqnarray*}[L_+,L_-]&=&i[L_y,L_x]-i[L_x,L_y]=\hbar(L_z+L_z)=2\hbar L_z \\ L^2&=&L_+L_-+L_z^2-\hbar L_z .\\ \end{eqnarray*}$$

* Example: What is the expectation value of $L_z$ in the state $\psi(\vec{r})=R(r)(\sqrt{2\over 3}Y_{11}(\theta,\phi)-i\sqrt{1\over 3}Y_{1-1}(\theta,\phi))$?*

* Example: What is the expectation value of $L_x$ in the state $\psi(\vec{r})=R(r)(\sqrt{2\over 3}Y_{11}(\theta,\phi)-\sqrt{1\over 3}Y_{10}(\theta,\phi))$?*