The Uncertainty Principle

You can never know both an object's exact position and its exact momentum at the same time — the more precisely you pin down where something is, the less precisely you can know how it's moving, and the other way around. This isn't because our instruments are clumsy; it's a fundamental limit built into nature itself, and it only becomes noticeable for very small particles like electrons.

Δx · Δp ≥ ℏ / 2

Δx — position uncertainty (m): how spread out the electron's possible positions are
Δp — momentum uncertainty (kg·m/s): how spread out the electron's possible momentum values are
ℏ — reduced Planck constant (J·s): a tiny fixed number, about 1 × 10⁻³⁴, that sets nature's uncertainty limit

An electron is confined inside a box 1 × 10⁻¹⁰ meters wide — about the size of an atom. What is the smallest possible uncertainty in its momentum?

Δx = 1 × 10⁻¹⁰ m
ℏ = 1 × 10⁻³⁴ J·s

Δp ≥ ℏ / (2 · Δx)
Δp ≥ (1 × 10⁻³⁴ J·s) / (2 × 1 × 10⁻¹⁰ m)

Δp ≥ 5 × 10⁻²⁵ kg·m/s

The same electron is now squeezed into a smaller box, only 1 × 10⁻¹¹ meters wide — ten times tighter. What is the new minimum momentum uncertainty?

5 × 10⁻²⁶ kg·m/s
5 × 10⁻²⁴ kg·m/s
5 × 10⁻²³ kg·m/s

If a physicist manages to pin down an electron's position much more precisely, what must happen to the uncertainty in its momentum?

It must decrease
It stays exactly the same
It must increase