  The quantum theory of linearized perturbations of the gravitational field of
a Schwarzschild black hole is presented. The fundamental operators are seen to
be the perturbed Weyl scalars $\dot\Psi_0$ and $\dot\Psi_4$ associated with the
Newman-Penrose description of the classical theory. Formulae are obtained for
the expectation values of the modulus squared of these operators in the
Boulware, Unruh and Hartle-Hawking quantum states. Differences between the
renormalized expectation values of both $\bigl| \dot\Psi_0 \bigr|^2$ and
$\bigl| \dot\Psi_4 \bigr|^2$ in the three quantum states are evaluated
numerically.
