  Quantum metrology enhances the sensitivity of parameter estimation using the
distinctive resources of quantum mechanics such as entanglement. It has been
shown that the precision of estimating an overall multiplicative factor of a
Hamiltonian can be increased to exceed the classical limit, yet little is known
about estimating a general Hamiltonian parameter. In this paper, we study this
problem in detail. We find that the scaling of the estimation precision with
the number of systems can always be optimized to the Heisenberg limit, while
the time scaling can be quite different from that of estimating an overall
multiplicative factor. We derive the generator of local parameter translation
on the unitary evolution operator of the Hamiltonian, and use it to evaluate
the estimation precision of the parameter and establish a general upper bound
on the quantum Fisher information. The results indicate that the quantum Fisher
information generally can be divided into two parts: one is quadratic in time,
while the other oscillates with time. When the eigenvalues of the Hamiltonian
do not depend on the parameter, the quadratic term vanishes, and the quantum
Fisher information will be bounded in this case. To illustrate the results, we
give an example of estimating a parameter of a magnetic field by measuring a
spin-$\frac{1}{2}$ particle, and compare the results for estimating the
amplitude and the direction of the magnetic field.
