 Solution

Introduce the dimensionless width $s=\hslash \alpha /(m\omega )>0$. Multiplying the variances givesThe minimum holds at all times precisely when $\alpha =m\omega /\hslash$, the oscillator ground-state width. For any other $\alpha$, it is still attained at the isolated quarter-period times $t=n\pi /(2\omega )$, including zero. At a fixed generic time, minimizing over $\alpha$ again selects $s=1$.

The excess product has a covariance interpretation:Thus the breathing Gaussian state in a harmonic oscillator saturates the Robertson-Schrödinger uncertainty relation throughout, even when its position-momentum product exceeds the lower bound in the Heisenberg uncertainty principle.