Ehrenfest’s Theorem

How do quantum averages move in time, and when does that motion look classical? Ehrenfest's theorem answers both questions with a single, exact equation.

The general equation of motion for an expectation value

Let $\hat{A}$ be an observable and $|\psi(t)\rangle$ a state evolving under the Schrödinger equation. The expectation value is $\langle \hat{A}\rangle = \langle\psi(t)|\hat{A}|\psi(t)\rangle$ . Differentiating with the product rule and using $i\hbar\,\partial_t|\psi\rangle = \hat{H}|\psi\rangle$ (and its conjugate) gives

\frac{d}{dt}\langle \hat{A}\rangle = \frac{i}{\hbar}\,\big\langle [\hat{H}, \hat{A}] \big\rangle + \Big\langle \frac{\partial \hat{A}}{\partial t} \Big\rangle.

This is Ehrenfest's theorem in its most general form. The first term comes from the state evolving; the second is present only if the operator itself carries explicit time dependence. For an observable with no explicit time dependence the second term vanishes and

\frac{d}{dt}\langle \hat{A}\rangle = \frac{i}{\hbar}\,\big\langle [\hat{H}, \hat{A}] \big\rangle.

The classical-looking pair

For a particle of mass $m$ in a potential $V(\hat x)$ , with $\hat H = \tfrac{\hat p^2}{2m} + V(\hat x)$ , the theorem produces

\frac{d}{dt}\langle \hat x\rangle = \frac{\langle \hat p\rangle}{m}, \qquad \frac{d}{dt}\langle \hat p\rangle = -\Big\langle \frac{\partial V}{\partial x}\Big\rangle.

These are derived purely from the canonical commutator $[\hat x, \hat p] = i\hbar$ . They look exactly like Newton's equations $\dot x = p/m$ and $\dot p = -V'(x)$ , but with every quantity replaced by its quantum average. This is the precise sense in which "averages obey classical mechanics."

Why it is not quite Newton's law

The second equation contains $\langle V'(\hat x)\rangle$ , the average of the force, not $V'(\langle \hat x\rangle)$ , the force evaluated at the average position. These differ unless $V$ is at most quadratic in $x$ (free particle, uniform field, harmonic oscillator), where $V'$ is linear and the average of a linear function equals the function of the average. For a general potential the spread of the wavepacket feels different parts of the force, so the centroid does not follow the classical trajectory exactly. Ehrenfest's theorem is exact for the averages, but it reduces to true classical dynamics only in the limit of narrow wavepackets in slowly varying potentials.

Conservation as a special case

If $\hat A$ has no explicit time dependence and commutes with the Hamiltonian, $[\hat H, \hat A] = 0$ , then $\tfrac{d}{dt}\langle \hat A\rangle = 0$ : its expectation value is conserved. Energy itself is the prototype, since $[\hat H, \hat H] = 0$ always gives $\tfrac{d}{dt}\langle \hat H\rangle = 0$ for a time-independent Hamiltonian. The next lesson develops this conservation idea in full.

The takeaway

Ehrenfest's theorem, $\tfrac{d}{dt}\langle\hat A\rangle = \tfrac{i}{\hbar}\langle[\hat H,\hat A]\rangle + \langle\partial_t \hat A\rangle$ , is the master equation for the time dependence of any expectation value. It explains the classical limit (averages obey Newton-like equations), pinpoints why that limit is only approximate (force of the average versus average of the force), and contains conservation laws as the case where an observable commutes with $\hat H$ .