  Suppose that the Hamiltonian acting on a quantum system is unknown and one
wants to determine what is the Hamiltonian. We show that in general this
requires a time $\Delta t$ which obeys the uncertainty relation $\Delta t
\Delta H \gtrsim 1$ where $\Delta H$ is a measure of how accurately the unknown
Hamiltonian must be estimated. We then apply this result to the problem of
measuring the energy of an unknown quantum state. It has been previously shown
that if the Hamiltonian is known, then the energy can in principle be measured
in an arbitrarily short time. On the other hand we show that if the Hamiltonian
is not known then an energy measurement necessarily takes a minimum time
$\Delta t$ which obeys the uncertainty relation $\Delta t \Delta E \gtrsim 1$
where $\Delta E$ is the precision of the energy measurement. Several examples
are studied to address the question of whether it is possible to saturate these
uncertainty relations. Their interpretation is discussed in detail.
