Position space
|ψ(q,t)|² on its energy level● classical particle
Phase space
1σ 2σ ellipseshading: Husimi
Angular width
exact σq semiclassical √Δqqlog scale
Advanced Quantum Mechanics · Tutorial 2b
A Gaussian packet in V(q) = −λ cos q, evolved exactly with the Schrödinger equation and compared with the semiclassical picture: the classical trajectory carrying a covariance ellipse. Pick an energy regime, then vary the packet and see when the ellipse stops describing it.
Position space. The density |ψ|² sits on the line E = H(x0, k0) above the potential; dotted parts of that line are classically forbidden. The two ends of the axis are the same point of the ring.
Phase space. Grey lines are classical orbits and the coloured curve is the separatrix. The shading is the exact Husimi function, the quantum state smoothed over one coherent-state cell, so it always looks a little wider than the ellipse.
Ellipse. The covariance Δ(t) from the linearized equations, carried by the classical trajectory. The classical swarm shows the full nonlinear classical flow of the same initial Gaussian.