Localization length
= Localization length
{title2=$|\psi_n|\sim e^{-|n-n_0|a/\xi}$}
The <localization length> $\xi$ measures the decay scale of a <localized eigenstate>'s <wave amplitude> envelope. Here $a$ is the site spacing and $n_0$ its centre. With this amplitude convention, probability decays on half that scale:
$$
|\psi_n|\sim e^{-|n-n_0|a/\xi},\qquad |\psi_n|^2\sim e^{-2|n-n_0|a/\xi}.
$$
Individual random tails fluctuate; this notation describes their typical envelope, not exact equality at every site. A <localization length> much greater than a finite sample's <length> makes a state look extended inside that sample. The <participation ratio> below is an effective site count, not an independent fitted value of $\xi$.