Position and momentum

An idea that clashes with common sense

In classical physics, we assume that a particle has, at every instant, a perfectly defined position and velocity. Good enough instruments would suffice to measure them as precisely as we like. Quantum mechanics claims the opposite, and not because of a technical limitation: it is impossible in principle.

The uncertainty principle, formulated by Werner Heisenberg in 1927, states:

One cannot know simultaneously, with perfect precision, the position and the momentum of a particle. The more precise one is, the fuzzier the other becomes.

The formulation

We write Δx for the uncertainty on the position and Δp for the uncertainty on the momentum (p = mv). The principle is written:

   Δx × Δp  ≥  ħ / 2

where ħ ("h-bar") is the reduced Planck constant, ħ = h / (2π) ≈ 1,05 × 10^-34 J·s.

The right-hand side is a constant, tiny but non-zero. That is what changes everything.

Reading the inequality as a trade-off

The product Δx × Δp can never drop below ħ/2. Since it is a product, reducing one factor forces the other to increase:

   position tres precise      position tres floue
   Δx petit                   Δx grand
   -----------|-----          --|--|--|--|--|--
        pour compenser :           pour compenser :
   Δp GRAND (vitesse floue)   Δp petit (vitesse precise)

   ... mais toujours :  Δx × Δp ≥ ħ/2

• If you localize the particle perfectly (Δx -> 0), then Δp must tend to infinity: its velocity becomes completely undetermined.
• If you know its velocity perfectly (Δp -> 0), then Δx blows up: it could be anywhere.

There is no state in which both are simultaneously zero. It is a trade-off imposed by nature.

Why we never notice it in everyday life

The factor ħ/2 is of the order of 10^-34. For a macroscopic object, this limit is so small that it is completely imperceptible. Take a 1 g marble whose position is known to within a micron:

   Δp ≥ ħ / (2 Δx) ≈ 10^-34 / (2 × 10^-6) ≈ 5 × 10^-29 kg·m/s
   soit Δv ≈ Δp / m ≈ 5 × 10^-26 m/s

An uncertainty on the velocity of 10^-26 m/s: perfectly unobservable. At our scale, position and velocity therefore seem perfectly defined. The uncertainty principle has visible consequences only at the scale of particles, where the mass is tiny.

In summary

Heisenberg's uncertainty principle imposes Δx × Δp ≥ ħ/2: one cannot make both the position and the momentum of a particle as precise as one wants. It is a fundamental trade-off, not a measurement flaw. Its value (~10^-34) makes it invisible for ordinary objects, but decisive at the atomic scale.