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State
Gaussian
Spread Δx
Spread Δp
Δx · Δp
Bound
minimum
Δx · Δp (units of ℏ)
ℏ/2 floor1ℏ2ℏ3ℏ
= ℏ/2
a Gaussian sits exactly on the floor
Heisenberg · Kennard1927

The trade is law

Δx·Δp ≥ ℏ/2. Sharpen where and you blur how fast, by an amount nature fixes exactly. Not a limit of instruments — a property of waves.

Fourier · the real reason

Two sides of one wave

Position and momentum are a Fourier pair. A pulse narrow in time is broad in frequency; narrow in space is broad in momentum. Same maths, everywhere.

Energy · timeΔE·Δt

Borrowed energy

The same law links energy and time: ΔE·Δt ≥ ℏ/2. It lets virtual particles flicker into being and sets the width of every spectral line.

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