energy-time uncertainty

The energy-time uncertainty relation, written ΔEΔt ≥ ℏ/2, links how sharply a system's energy is defined to how long it persists or how fast it changes. A state that lives for only a brief moment cannot have a perfectly sharp energy; a state with a precisely defined energy must, in turn, be unchanging in time. The shorter the lifetime, the broader the energy must be smeared.

This relation looks just like the position-momentum one, but it must be read more carefully, because in ordinary quantum mechanics time is not an observable with its own operator the way position is. Here Δt is best understood as a characteristic timescale — how long a state takes to change appreciably — rather than an uncertainty in a measured clock reading. With that reading, the relation becomes a precise and useful statement about how quickly a quantum system can evolve.

Its consequences are everywhere in physics. An unstable particle or excited atomic state that decays quickly has a correspondingly broad spread of energies, seen directly as the natural linewidth of spectral lines: short-lived states give fuzzy, wide lines, long-lived ones give sharp, narrow ones. The same relation explains why very short laser pulses contain a wide band of frequencies, a fact at the heart of ultrafast optics.