Gaussian wave packet

A Gaussian wave packet is a wavefunction whose spatial amplitude is a Gaussian function. Free quantum evolution preserves its Gaussian form while changing its complex width and spreading its probability density.

Under a quantum harmonic oscillator Hamiltonian, the Heisenberg equations of motion giveSubtract the expectation values, square the first equation, and use the symmetrized covariance $C_0=⟨\{x-⟨x⟩,p-⟨p⟩\}⟩/2$. This yieldsThese moment equations hold for any state with finite second moments; a quadratic Hamiltonian additionally preserves Gaussian wave packets. A coherent state has ${\sigma }_x(0{)}^2=\hslash /(2m\omega )$, ${\sigma }_p(0{)}^2=m\hslash \omega /2$, 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 $2\omega$.

Let the initial Gaussian wave packet beFourier transforming, multiplying each momentum amplitude by $e^{-i{p}^{2}t/(2m\hslash )}$ and evaluating the Gaussian integral givesIts probability density is a normalized Gaussian with center $x_0+p_{0}t/m$ and width ${\sigma }_x={\sigma }_0\sqrt{1+{\eta }^2}$; the momentum width remains ${\sigma }_p=\hslash /(2{\sigma }_0)$. Its symmetrized position-momentum covariance is $C_{xp}=t{\sigma }_p^2/m$, soThus free spreading preserves equality in the Robertson-Schrödinger uncertainty relation while producing a nonzero quadratic position-dependent phase and a product ${\sigma }_x{\sigma }_p$ larger than $\hslash /2$. More generally, ${\sigma }_x(t{)}^2={\sigma }_x(0{)}^2+2t{C}_{xp}(0)/m+t^2{\sigma }_p(0{)}^2/m^2$, so an initially correlated Gaussian can first contract.

For the Hamiltonian $H=(p^2-x^2)/2$, a Gaussian wave packet $A{e}^{-B{x}^2}$ satisfies $\dot{A}=-i\hslash AB$ and $\dot{B}=-i/(2\hslash )-2i\hslash B^2$. The displayed solution is normalizable for real $\varphi$ exactly when $\sin (2\varphi )>0$; its second moments of a complex Gaussian wave packet areBoth grow as $\hslash e^{2t}/[4\sin (2\varphi )]$, while their difference and hence the mean energy remain constant.