  We present the first numerical computation of the neutral fermion gap,
$\Delta_\psi$, in the $\nu=5/2$ quantum Hall state, which is analogous to the
energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We
find $\Delta_\psi \approx 0.027 \frac{e^2}{\epsilon \ell_0}$, comparable to the
charge gap, and discuss the implications for topological quantum information
processing. We also deduce an effective Fermi velocity $v_F$ for neutral
fermions from the low-energy spectra for odd numbers of electrons, and thereby
obtain a correlation length $\xi_{\psi}={v_F}/\Delta_{\psi} \approx 1.3\,
\ell_0$. We comment on the implications of our results for electronic
mechanisms of superconductivity more generally.
