The uncertainty relation between energy and time of measurement

Contrary to a wide-spread impression, the possibility of measuring an energy in a finite time without changing its initial value $(E'=E_0)$ is not in contradiction with the principles of quantum mechanics. The relation $\Delta(E'-E_0)\Delta t\ge\hslash$ holds only in the case when the energy of interaction between the quantum system in question and the apparatus is a function of a coordinate of the system. The condition for a nonperturbing energy measurement is that the interaction energy $H_1$, of the system and the apparatus depend on the energy operator $\hat E$ and that the operators $\hat H$ and $\hat E$ commute. It is also possible to have a nonperturbing measurement in which the error in measuring the energy is so small that $\Delta E\ll\hslash/\Delta t$. Measurement of the energy of a given system is accompanied by an increase in the uncertainty $\Delta\varepsilon$ of the energy of the apparatus. The error $\Delta E$ in the measurement of the system's energy and the perturbation $\Delta\varepsilon$ of the energy of the apparatus are connected by the relations $(\Delta E+\Delta\varepsilon)\cdot\Delta t\ge\hslash$ and $\Delta E\cdot\Delta\varepsilon)\ge(\hslash/2\Delta t)^2$.