The energy-time uncertainty and the quantum vacuum

A second form of the principle

The position-momentum pair is not the only one concerned. There is an analogous uncertainty relation between energy and time:

   ΔE × Δt  ≥  ħ / 2

It reads as follows: a state that lasts only a very short time Δt does not have a perfectly defined energy — its energy is fuzzy by an amount ΔE. Only a state of infinite duration would have a perfectly sharp energy.

Consequence 1: the width of spectral lines

An atom in an excited state stays there only for an instant (Δt short) before falling back. According to ΔE × Δt ≥ ħ/2, the energy of this state is therefore slightly fuzzy (ΔE non-zero).

Result: spectral lines do not have perfect sharpness; they have a small width. And this width tells us about the lifetime of the state:

   etat de longue duree (Δt grand)  ->  ΔE petit  ->  raie tres fine
   etat de courte duree (Δt petit)  ->  ΔE grand  ->  raie plus large

Measuring the width of a line therefore means indirectly measuring the lifetime of the excited state. The temporal fuzziness is read in the spectrum.

Consequence 2: the vacuum is not empty

The most astonishing consequence concerns the vacuum. Classically, the vacuum is nothingness: no energy, nothing. But ΔE × Δt ≥ ħ/2 allows the energy to fluctuate for very brief durations. Over a short enough instant Δt, an energy ΔE can "appear" without violating the conservation of energy — provided it disappears immediately.

   vide "classique" :  ————————————  (rien, energie nulle et constante)

   vide quantique   :   .  ' .  ' ` .  '   <- fluctuations : des paires de
                       ` .  ' ` .  '  `       particules apparaissent et
                                              disparaissent en un instant Δt

These vacuum fluctuations cause pairs of "virtual" particles to be constantly born and annihilated. This is not speculation: their effects are measured (Casimir effect, Lamb shift in spectra, Hawking radiation of black holes). The quantum vacuum seethes permanently.

In summary

The relation ΔE × Δt ≥ ħ/2 links the energy of a state to its duration: a short-lived state has a fuzzy energy. This gives spectral lines a measurable width (linked to the lifetime of excited states) and allows the vacuum to fluctuate — hence pairs of virtual particles that appear and disappear, with very real physical effects.