Throwing a quantum particle - the free fall in quantum mechanics

The gravitational field near the earth's surface can be described by a constant and homogeneous force field acting in the vertical direction (i.e., by a linear potential). Here we see the motion of a quantum mechanical particle in two dimensions. The classical analogue is a mass point being thrown vertically.

The particle is initially (at t=0) described by a Gaussian wave packet. The initial momentum and initial position are chosen in such way that the classical turning point is at x=0.

"Visual Quantum Mechanics - Book One" explains that the exact solution of the problem uses the Avron Herbst formula to determine the time evolution. The average position follows classical mechanics (similar to the free motion and the motion in the field of a harmonic oscillator). The position of a corresponding classical particle is indicated by a white spot in the animation. At the classical turning point, the solution is (apart from a phase factor) just a Gaussian wave packet at rest, but spreaded according to the time needed to reach that point.