How does Ehrenfest's theorem relate classical and quantum mechanics in the context of a quantum state's evolution?

Ehrenfest’s theorem shows that the average position and momentum of a quantum particle follow Newton’s equations for a particle moving in a potential, just like a classical point particle. It does this by taking the expectation values of the quantum operators for position and momentum and differentiating them with respect to time, then using the Schrödinger equation. The resulting equations are identical to the classical equations of motion, so the quantum state’s evolution reproduces classical behavior on average. For example, a Gaussian wave packet in a harmonic potential will have its center move like a classical oscillator, even though the packet’s shape spreads over time. Thus Ehrenfest’s theorem bridges the two theories by demonstrating that quantum dynamics reduces to classical dynamics for the mean values of observables.