Uncertainty Principle Sandbox — σ_x · σ_p = ℏ/2 for the Gaussian

Squeeze the position wavepacket and the momentum spread explodes. The Gaussian is the unique minimum-uncertainty state: any other shape gives σ_x·σ_p > ℏ/2.

|ψ(x)|² — position space

|φ̃(p)|² — momentum space

a = 1.00 nm (Gaussian width parameter)

Wavepacket shape

σ_x = — nm × σ_p = — ℏ/nm = — ℏ
(floor = 0.500 ℏ)

Uncertainty Principle Sandbox — σx · σp = ℏ/2 for the Gaussian

Uncertainty Principle Sandbox — σ_x · σ_p = ℏ/2 for the Gaussian