Bessel second-moment identity
= Bessel second-moment identity
{c}
{title2=$\sum_{m\in\mathbb Z}m^2J_m(z)^2=z^2/2$}
For real $z$, the <probabilities> $J_m(z)^2$ over integer angular-momentum modes satisfy $\sum_m m^2J_m(z)^2=z^2/2$. Differentiate the unit-modulus function $e^{iz\cos\theta}$ and use Parseval to obtain the identity; its zero first moment follows from the symmetry of opposite orders.