  In the last several years, the Casimir energy for a variety of
1+1-dimensional integrable models has been determined from the exact S-matrix.
It is shown here how to modify the boundary conditions to project out the
lowest-energy state, which enables one to find excited-state energies. This is
done by calculating thermodynamic expectation values of operators which
generate discrete symmetries. This is demonstrated with a number of perturbed
conformal field theories, including the Ising model, the three-state Potts
model, ${\bf Z}_n$ parafermions, Toda minimal S-matrices, and massless
Goldstinos.
