Squeezed state of the two-dimensional harmonic oscillator

Here we see the time evolution of a wave packet in the field of a two-dimensional harmonic oscillator. The harmonic oscillator is the quantum analog of a mass point attached to a spring. There are many similarities between the classical and the quantum motion of a harmonic oscillator. This becomes only apparent if one considers not only the eigenstates (=states with sharp energies), but superpositions like the Gaussian function shown above. You can see that the quantum motion (like the classical oscillation) is periodic in time. After a certain time period the wave function returns to the initial condition. The initial function has average momentum zero and average initial position at some distance from the center of attraction. During its time evolution the wave packet starts spreading and gains speed. The density of the stripes are a measure for the momentum of the wave packet. The momentum is maximal (minimal wave length) when the particle moves through the origin. After passing the origin, the wave function contracts again until it reaches the turning point. For the harmonic oscillator (like for the free motion and the free fall in a linear potential) that the average position and momentum (the expectation values) follow precisely the laws of classical physics.

"Visual Quantum Mechanics - Book One" describes the motion of Gaussian wave packets in the field of a harmonic oscillator in full generality (including a derivation of the Mehler kernel). The CD contains several movies showing squeezed (Gaussian) states and coherent states in one and two dimensions.