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Ehrenfest Theorem

Also known as: Ehrenfest's Equations · Quantum-Classical Correspondence

Quantum averages obey Newton-like equations — the wavepacket's center moves classically.

d⟨x^⟩dt=⟨p^⟩m,d⟨p^⟩dt=−⟨∂V∂x⟩\frac{d\langle \hat{x}\rangle}{dt} = \frac{\langle \hat{p}\rangle}{m}, \qquad \frac{d\langle \hat{p}\rangle}{dt} = -\left\langle \frac{\partial V}{\partial x}\right\rangle
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A Gaussian packet sloshes in a harmonic well while a classical ball rides the same trajectory — the mean position ⟨x⟩ traces an exact classical cosine.

Equivalent forms

ddt⟨A^⟩=1iℏ⟨[A^,H^]⟩+⟨∂A^∂t⟩\frac{d}{dt}\langle \hat{A}\rangle = \frac{1}{i\hbar}\langle[\hat{A},\hat{H}]\rangle + \left\langle\frac{\partial \hat{A}}{\partial t}\right\rangle
md2⟨x^⟩dt2=⟨F(x)⟩m\frac{d^2\langle \hat{x}\rangle}{dt^2} = \langle F(x)\rangle
Newton's second law is the shadow that quantum mechanics casts on averages — exact for free particles, uniform fields, and harmonic oscillators.