Energy-time and frequency-time uncertainty relations: exact inequalities

We give a short review of known exact inequalities that can be interpreted as “energy-time” and “frequency-time” uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency-time uncertainty product. Also, we calculate the “stationarity time” for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are “more fragile” than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.