= Solution
A localized quantum particle is described by a <wave packet>: a coherent <quantum superposition> of <momentum> components, rather than a single delocalized <plane wave>. The <Born rule> assigns the <position> <probability density> $|\psi(x,t)|^2$. A useful normalized initial <Gaussian wave packet> is
$$
\psi(x,0)=(2\pi\sigma_0^2)^{-1/4}\exp\left[-\frac{(x-x_0)^2}{4\sigma_0^2}+\frac{ip_0(x-x_0)}\hbar\right].
$$
Its mean <position> and <momentum> are $x_0$ and $p_0$, and its <standard deviations> satisfy $\sigma_x(0)=\sigma_0$, $\sigma_p=\hbar/(2\sigma_0)$. This unchirped initial state saturates the <Heisenberg uncertainty relation>.
For a <free particle>, the momentum-space <Schrödinger equation> multiplies each <amplitude> by $e^{-ip^2t/(2m\hbar)}$. The inverse <Fourier transform> is a <Gaussian integral>, giving the <free Gaussian wave packet evolution>
$$
\psi(x,t)=\frac{(2\pi\sigma_0^2)^{-1/4}}{\sqrt{1+i\eta}}\exp\left[-\frac{(x-x_0-p_0t/m)^2}{4\sigma_0^2(1+i\eta)}+\frac{ip_0(x-x_0)}\hbar-\frac{ip_0^2t}{2m\hbar}\right],\qquad \eta=\frac{\hbar t}{2m\sigma_0^2}.
$$
One way to obtain this expression explicitly is to write the initial <momentum> <amplitude> as proportional to $\exp[-\sigma_0^2(p-p_0)^2/\hbar^2-ipx_0/\hbar]$, multiply by the free evolution phase and complete the square in $p$. Taking the modulus squared removes the position-dependent phase and leaves
$$
\boxed{|\psi(x,t)|^2=\frac{1}{\sqrt{2\pi}\sigma_x(t)}\exp\left[-\frac{(x-x_c(t))^2}{2\sigma_x(t)^2}\right],\quad x_c(t)=x_0+\frac{p_0t}{m},\quad \sigma_x(t)=\sigma_0\sqrt{1+\left(\frac{t}{\tau}\right)^2},\quad \tau=\frac{2m\sigma_0^2}{\hbar}.}
$$
The <expectation value> follows uniform classical motion while the density spreads. A narrower initial packet needs a wider <momentum> distribution and therefore spreads more rapidly. At long times its width grows linearly with $|t|$; its <variance> grows quadratically, unlike the linear <variance> growth of ordinary <diffusion>. The <probability continuity equation> preserves total <probability> despite the decreasing peak height. The mean <momentum> and the mean <energy> $(p_0^2+\sigma_p^2)/(2m)$ remain constant, since the <momentum> populations do not change.
The same result has a wave interpretation. For a free nonrelativistic particle $\omega(k)=\hbar k^2/(2m)$, so the <group velocity> is $d\omega/dk=\hbar k/m$, equal to $p_0/m$ at the packet center. The curvature $d^2\omega/dk^2=\hbar/m$ makes different <momentum> components travel with different <group velocities>, producing dispersion. The <phase velocity> is $\omega/k=\hbar k/(2m)$ for this <energy> convention and does not describe the motion of the envelope. The evolving quadratic spatial phase is essential to the spreading; a density snapshot alone does not specify the full <quantum state>.
A freely spreading <Gaussian wave packet> remains a pure Gaussian state but develops position-momentum correlations. Its symmetrized <covariance> is $C_{xp}=\langle\{x-\langle x\rangle,p-\langle p\rangle\}\rangle/2=t\sigma_p^2/m$. Consequently
$$
\sigma_x\sigma_p=\frac\hbar2\sqrt{1+\eta^2},\qquad \boxed{\sigma_x^2\sigma_p^2-C_{xp}^2=\frac{\hbar^2}{4}.}
$$
Equality in the <Robertson-Schrödinger uncertainty relation> survives at every time, whereas the simpler product $\sigma_x\sigma_p$ equals $\hbar/2$ only initially. A <chirped Gaussian wave packet> can already have $C_{xp}(0)\ne0$. More generally, $x(t)=x(0)+p(0)t/m$ gives $\sigma_x(t)^2=\sigma_x(0)^2+2tC_{xp}(0)/m+t^2\sigma_p(0)^2/m^2$. A sufficiently negative initial <covariance> permits initial contraction before subsequent spreading.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-1-gaussian-packets.png]
{title=Free Gaussian spreading, constant coherent-state width and the breathing width of a squeezed state}
A <potential energy> changes this behavior. A quadratic potential preserves the Gaussian form because the evolution of <position> and <momentum> is linear. For a <quantum harmonic oscillator>, the <Gaussian covariance in a harmonic potential> is
$$
\sigma_x(t)^2=\sigma_x(0)^2\cos^2\omega t+\frac{\sigma_p(0)^2}{m^2\omega^2}\sin^2\omega t+\frac{2C_{xp}(0)}{m\omega}\sin\omega t\cos\omega t.
$$
A <coherent state> has $\sigma_x^2=\hbar/(2m\omega)$, $\sigma_p^2=m\hbar\omega/2$ and zero centered <covariance>. Its width is constant, and its center follows $x_c(t)=x_0\cos\omega t+p_0\sin\omega t/(m\omega)$. In oscillator notation its <coherent state> label rotates as $\alpha(t)=\alpha(0)e^{-i\omega t}$, apart from the common ground-energy phase of the <state vector>. A <squeezed coherent state> generally has a width that breathes at $2\omega$. Thus being Gaussian does not imply either free spreading or constant width; the <Hamiltonian operator> and initial <covariance> determine the result.
For a general nonquadratic potential, different parts of the packet experience different nonlinear forces, and an initially Gaussian profile need not remain Gaussian. <Ehrenfest theorem> gives the exact relations $d\langle x\rangle/dt=\langle p\rangle/m$ and $d\langle p\rangle/dt=-\langle V'(x)\rangle$. Replacing the last mean by $-V'(\langle x\rangle)$ is exact for a quadratic potential and approximate for a packet narrow compared with the scale on which the force varies. This connects packet motion to the <correspondence principle> without treating a <quantum state> as a definite classical path. Scattering can split a packet into reflected and transmitted components, while confinement can produce oscillations and interference between discrete <energy eigenstates>. Their relative phases evolve even when their populations remain constant. Localization, spreading and coherent motion all follow from the same linear <unitary time evolution>.
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