= Gaussian covariance in a harmonic potential
{c}
Under a <quantum harmonic oscillator> Hamiltonian, the <Heisenberg equations of motion> give
$$
x(t)=x(0)\cos\omega t+\frac{p(0)}{m\omega}\sin\omega t,\qquad p(t)=p(0)\cos\omega t-m\omega x(0)\sin\omega t.
$$
Subtract the <expectation values>, square the first equation, and use the symmetrized <covariance> $C_0=\langle\{x-\langle x\rangle,p-\langle p\rangle\}\rangle/2$. This yields
$$
\sigma_x(t)^2=\sigma_x(0)^2\cos^2\omega t+\frac{\sigma_p(0)^2}{m^2\omega^2}\sin^2\omega t+\frac{2C_0}{m\omega}\sin\omega t\cos\omega t.
$$
These moment equations hold for any state with finite second moments; a quadratic Hamiltonian additionally preserves <Gaussian wave packets>. A <coherent state> has $\sigma_x(0)^2=\hbar/(2m\omega)$, $\sigma_p(0)^2=m\hbar\omega/2$, and $C_0=0$, so its width stays fixed while its center follows the classical orbit. A <squeezed coherent state> generally has a width that oscillates at $2\omega$.
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