= Angular momentum raising operator
{title2=$L_+=L_1+iL_2$}
= Raising operator for angular momentum
{synonym}
The <angular momentum raising operator> $L_+=L_1+iL_2$ satisfies $[L_3,L_+]=\hbar L_+$ 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 $L_-L_+=L^2-L_3^2-\hbar L_3$, hence
$$
\|L_+\psi\|^2=\hbar^2[\lambda-m(m+1)]\|\psi\|^2
$$
for simultaneous <eigenvalues> $\lambda\hbar^2,m\hbar$. For normalized $|l,m\rangle$ this yields $L_+|l,m\rangle=\hbar\sqrt{(l-m)(l+m+1)}|l,m+1\rangle$ 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 $L_2+iL_3$ raises the $L_1$ projection.
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