The Coherent State

A free wavepacket spreads as it travels, but the harmonic oscillator has one family of states that never do: the coherent states. Displace the ground state of the oscillator and let it go, and you get a Gaussian that keeps the exact ground-state width $\sigma_0=\sqrt{\hbar/2m\omega}$ for all time while its centre slides back and forth along the classical trajectory, $\langle x\rangle(t)=x_0\cos\omega t$. It is the closest a quantum state comes to a classical particle: a minimum-uncertainty blob that oscillates in the well without ever blurring. The scene shows the parabolic potential, the energy level, and the live packet $|\psi|^2$ riding on it, with the real part of the wavefunction wiggling fastest as the packet whips through the centre (where its momentum is largest) and going smooth at the turning points (where it stops). The lower panel plots the phase-space point $(\langle x\rangle,\langle p\rangle)$ tracing its energy ellipse, the same closed orbit a classical oscillator follows, and splits the energy into kinetic and potential parts that trade off as the packet swings while their sum stays fixed. Raise the amplitude $x_0$ and the packet becomes small against its swing, the classical limit; raise the frequency $\omega$ and the well stiffens, squeezing the packet narrower.