Ehrenfest theorem

Ehrenfest's theorem says that the average values of quantum quantities evolve in time according to equations that mirror the laws of classical mechanics. If you track the expectation value of a particle's position and the expectation value of its momentum, you find that the average position changes with the average momentum, and the average momentum changes with the average force — Newton's laws, but for averages rather than for sharp trajectories.

This is a reassuring bridge between the quantum and classical pictures. Quantum mechanics replaces a definite path with a spread-out wavefunction, yet Ehrenfest shows that the centre of that spread tends to move much as a classical particle would. It is why a wave packet, on the whole, glides along a trajectory close to the one a little ball would follow under the same forces, at least while the packet stays compact.

The agreement is, however, only approximate and worth stating honestly. The average force depends on the force evaluated across the whole spread of the wavefunction, not on the force at the average position. When the force varies smoothly over the packet's width, the two nearly coincide and classical-looking motion emerges; when the force changes sharply within the packet, the average motion can depart from any classical path. Ehrenfest's theorem thus explains why classical mechanics works so well, and also hints at the conditions under which it must fail.