  Two methods are proposed to evaluate the second-order elastic constants of
quantum mechanically treated solids. One method is based on path-integral
simulations in the (NVT) ensemble using an estimator for elastic constants. The
other method is based on simulations in the (NpT) ensemble exploiting the
relationship between strain fluctuations and elastic constants. The strengths
and weaknesses of the methods are discussed thoroughly. We show how one can
reduce statistical and systematic errors associated with so-called primitive
estimators. The methods are then applied to solid argon at atmospheric
pressures and solid helium 3 (hcp, fcc, and bcc) under varying pressures. Good
agreement with available experimental data on elastic constants is found for
helium 3. Predictions are made for the thermal expectation value of the kinetic
energy of solid helium 3.
