The quantum-mechanical framework in which observables are associated with Hermitian
operators is too narrow to discuss measurements of such important physical quantities
as elapsed time or harmonic-oscillator phase. We introduce a broader framework that
allows us to derive quantum-mechanical limits on the precision to which a parameter---e.g.,
elapsed time---may be determined via arbitrary data analysis of arbitrary measurements
on \(N\) identically prepared quantum systems. The limits are expressed as generalized
Mandelstam-Tamm uncertainty relations, which involve the operator that generates displacements
of the parameter---e.g., the Hamiltonian operator in the case of elapsed time. This
approach avoids entirely the problem of associating a Hermitian operator with the
parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty
relations for spatial displacement and momentum, harmonic-oscillator phase and number
of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations
involving the displacement and Lorentz-rotation parameters of the Poincar\'e group.