= Breathing Gaussian state in a harmonic oscillator
{title2=$\Sigma(t)=M(t)\Sigma(0)M(t)^{\mathsf T}$}
A real centered <Gaussian wave packet> with initial <position> <variance> $1/(2\alpha)$ and <momentum> <variance> $\hbar^2\alpha/2$ evolves under a <quantum harmonic oscillator> by
$$
M(t)=\begin{pmatrix}\cos\omega t&\sin\omega t/(m\omega)\\
-m\omega\sin\omega t&\cos\omega t\end{pmatrix}.
$$
The <covariance matrix> obeys $\Sigma(t)=M\Sigma(0)M^{\mathsf T}$ and $\det\Sigma=\hbar^2/4$. Thus the <Robertson-Schrödinger uncertainty relation> stays saturated, although the position-momentum <variance> product can exceed $\hbar^2/4$ because of nonzero <quantum covariance>. The widths are stationary only for $\alpha=m\omega/\hbar$; otherwise they breathe with <frequency> $2\omega$.
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