Uncertainty Minimizer — σx · σp ≥ ħ/2

Only the Gaussian saturates the Heisenberg bound (ratio = 1.000 exactly) — all other shapes give ratio > 1

1.20 (ħ=1)
ψ(x) real part
|φ(k)|² (scaled)
Uncertainty Gauge — ratio × (ħ/2)
Dashed line = minimum ħ/2 (Gaussian only)
Quantum Statistics
Position spread σx
—
ħ=1 units
Momentum spread σp
—
ħ=1 units
Product σx · σp
—
in units of ħ/2
Ratio = (σx·σp) / (ħ/2)
—
—
Heisenberg bound: σx · σp ≥ ħ/2
Minimum-uncertainty state: Gaussian only.
Ratio ≡ 1.000 ← saturated.
Any other shape → ratio > 1.