 Angular momentum raising operator

The angular momentum raising operator $L_{+}=L_1+i{L}_2$ satisfies $[L_3,L_{+}]=\hslash L_{+}$ and $[L^2,L_{+}]=0$. It preserves the total angular momentum while increasing the third-component magnetic quantum number by one, whenever its result is nonzero. Its adjoint operator is the angular momentum lowering operator $L_{-}$. Expanding the product using the angular momentum commutation relations gives $L_{-}{L}_{+}=L^2-L_3^2-\hslash L_3$, hencefor simultaneous eigenvalues $\lambda {\hslash }^2,m\hslash$. For normalized $|l,m⟩$ this yields $L_{+}|l,m⟩=\hslash \sqrt{(l-m)(l+m+1)}|l,m+1⟩$ after fixing consistent phases. At $m=l$ the result is zero, the upper angular momentum ladder endpoint. For a different quantization axis, cyclically permute the components; for example $L_2+i{L}_3$ raises the $L_1$ projection.