minimum-uncertainty state
A minimum-uncertainty state is a quantum state that hits the uncertainty bound exactly, turning the inequality ΔxΔp ≥ ℏ/2 into an equality ΔxΔp = ℏ/2. Such a state is as close to having a definite position and momentum together as the laws of quantum mechanics ever allow. It represents the sharpest possible joint knowledge of a pair of conjugate quantities.
For position and momentum, the state that achieves this is a Gaussian — a smooth bell-shaped wavefunction. There is something elegant here: the Gaussian is the unique shape whose Fourier transform is also a Gaussian, so it manages to be reasonably narrow in both position and momentum at the same time. Any other shape wastes a little, spreading wider than the strict minimum in one variable or the other.
Minimum-uncertainty states are not just mathematical curiosities. The ground state of a harmonic oscillator is a Gaussian and therefore a minimum-uncertainty state, which is why it sits at the lowest energy allowed rather than dead still. Coherent states of light, the closest quantum description of a steady laser beam, are also minimum-uncertainty states, which is part of why a laser behaves so nearly like an ideal classical wave.
A Gaussian wavefunction is the unique shape that meets the uncertainty bound with equality.
Saturating the bound for position and momentum does not mean both are well known in absolute terms — it only means their product is as small as allowed. A squeezed state can push Δx below the Gaussian value, but only by paying with a larger Δp.