In quantum mechanics, how does the concept of operators relate to the uncertainty principle, specifically in measuring the energy of a particle?

In quantum mechanics, an operator is a mathematical tool that tells us how to measure a physical quantity, like energy. The energy operator (Hamiltonian) does not commute with the time operator, meaning that measuring energy precisely changes our knowledge of the particle’s time, and vice versa. This non‑commutation leads directly to the uncertainty principle: the more accurately we know a particle’s energy, the less precisely we can know the time at which that energy was measured. For example, if a particle’s energy is pinned down to a very narrow value, the wave packet’s phase evolves slowly, making the exact instant of measurement fuzzy. Thus, operators and their commutation relations explain why we cannot simultaneously know a particle’s exact energy and the exact time of that measurement.