Measuring energy, estimating Hamiltonians, and the time-energy uncertainty relation

Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\Delta t$ that 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 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 with arbitrarily large precision 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.