Animated Gaussian wavefunction of a coherent state |α⟩ in a 1-D harmonic well: rigid σ = 1/√2 packet whose centroid traces the classical orbit ⟨x⟩(t) = √2|α|cos(ωt − φ_α). Side-by-side phase space, |ψ(x,t)|², and ⟨x⟩(t) trace.
About this model
This lab animates a coherent state |α⟩ in a one-dimensional quantum harmonic oscillator. The wavefunction is a Gaussian of fixed width σ = 1/√2 (in oscillator units ħ = m = ω = 1) whose centroid follows the classical orbit ⟨x⟩(t) = √2 |α| cos(ωt − φ_α). Side-by-side views show the phase-space point, the probability density |ψ(x,t)|², and the time trace of ⟨x⟩(t). The model assumes a pure coherent state in an ideal harmonic well: no anharmonicity, no decoherence, and no squeezing. You vary |α| and the phase φ_α to change orbit radius and initial phase, and watch time evolution to see that the packet shape does not disperse while the mean follows classical harmonic motion.
Who it's for: Undergraduate and early graduate quantum mechanics and quantum optics courses introducing coherent states and the classical limit of the oscillator.
Key terms
Coherent state
Harmonic oscillator
Gaussian wave packet
Expectation value
Classical orbit
Phase space
How it works
A coherent state |α⟩ of the quantum harmonic oscillator: a minimum-uncertainty Gaussian whose centroid follows the classical orbit ⟨x⟩(t) = √2|α| cos(ωt − φ_α). Watch the wavefunction in real space, the rotating point in phase space, and the ⟨x⟩(t) trace simultaneously.
Frequently asked questions
Why does a coherent-state packet not spread like a free Gaussian?
In a harmonic oscillator the energy spacing is constant, so the relative phases of number components rephase every period. A coherent state is a minimum-uncertainty Gaussian matched to the oscillator ground-state width; the restoring potential keeps σ fixed at 1/√2 while the centroid oscillates. Free-space packets lack that potential and disperse.
Is ⟨x⟩(t) exactly classical?
For a coherent state in a purely harmonic well, Ehrenfest’s theorem gives closed equations for ⟨x⟩ and ⟨p⟩ identical to Newton’s laws, so ⟨x⟩(t) = √2 |α| cos(ωt − φ_α) matches the classical orbit. Anharmonic corrections would make quantum averages deviate from a single classical trajectory.
What if |α| is very small?
As |α| → 0 the state approaches the ground state |0⟩: the centroid sits at the origin and |ψ|² is the vacuum Gaussian. Finite |α| simply displaces that same-shaped packet along a larger classical ellipse; there is no threshold where quantum motion suddenly becomes classical.