Heisenberg uncertainty principle explained
The tighter you pin a particle to one point, the more its momentum blurs. Call the spread of position Δx and the spread of momentum Δp, and they are a seesaw where shrinking one side grows the other. For a Gaussian packet the product falls all the way to the floor Δx·Δp = ℏ/2, and it cannot go below that. The first picture starts from one small hump in space.
A particle with a sharp position is one small hump in space. Draw the probability density |ψ(x)|² of a Gaussian packet and change the width. Shrink the width and the hump grows narrow and tall, and widen it and the hump grows low and broad. The area stays fixed at 1, so a narrower hump has to grow taller. Push the width toward 0 to make a perfect point and the hump shoots up without limit, which is physically impossible, and that is the first sign that position cannot be narrowed without end.
The same width sets the position spread and the momentum spread together. For a Gaussian packet the position spread is proportional to the width, and the momentum spread is proportional to 1/σ, namely ℏ/(2σ). Shrink the width and the position bar shrinks while the momentum bar rises by as much, and the product of the two bars stays at the same value. One width setting both spreads is because the two are a pair from one root, the same trade as a shorter sound having a blurrier pitch.
Write the limit in one line and it is the Heisenberg relation Δx·Δp ≥ ℏ/2. Here ℏ = h/2π is the reduced Planck constant. Draw the boundary Δp = ℏ/(2Δx) and shade below it, and a point may sit on the curve or above it but never enters the shaded region below. A Gaussian packet lies exactly on that boundary. This floor is not clumsy measurement. Nature set it, and it is also why an electron does not fall all the way into the nucleus and why an atom does not collapse.
Why does narrowing blur the momentum. A hump with a sharp position is built by adding plane waves of many wavenumbers. Raise the count N of waves you add. With only one there is an endless repeating wave and no position, and as you add more they reinforce only near x = 0 and the hump grows sharp. A tight confinement needs that many wavenumbers. Those added wavenumbers being spread wide means the momentum is spread wide, so the blur of momentum arrives at the very moment the narrow hump is made.
A description in position and a description in momentum are two languages for the same particle, and they are paired by a Fourier transform. The left side is a Gaussian in position and the right side is a Gaussian in momentum, and their widths are reciprocals. Shrink the position width and the left side narrows while the right side widens as σp = 1/(2σx). Making one side sharp blurs the other because of this reciprocal relation. A Fourier pair whose widths are reciprocals belongs to every wave, not only to the quantum, and because nature writes even a particle as a wave the uncertainty became a law of matter.