Ehrenfest Theorem Formulas

Formula

$$\frac{d}{dt}\langle A\rangle =\frac{i}{\hbar}\langle[H,A]\rangle +\left\langle\frac{\partial A}{\partial t}\right\rangle.$$

For $H=p^2/(2m)+V(x)$,

$$\frac{d}{dt}\langle x\rangle=\frac{\langle p\rangle}{m}, \qquad \frac{d}{dt}\langle p\rangle=-\langle V'(x)\rangle,$$

so

$$m\frac{d^2}{dt^2}\langle x\rangle=-\langle V'(x)\rangle.$$

For $q=\langle x\rangle$ and variance $\sigma_x^2$,

$$\langle V'(x)\rangle =V'(q)+\frac12V'''(q)\sigma_x^2+\cdots.$$

Assumptions and Conventions

• Schrödinger and Heisenberg pictures give the same expectation-value law.
• The explicit partial derivative is included when $A$ depends on time.
• States and operator products must lie in domains that justify the commutator and differentiation steps.

Symbols

Symbol	Meaning
$q$	mean position
$\sigma_x^2$	position variance
$V'(x)$	force-gradient expression; force is $-V'(x)$
$[H,A]$	commutator $HA-AH$

Validity and Warnings

• In general $\langle V'(x)\rangle\ne V'(\langle x\rangle)$.
• Exact closure of the first moments occurs for potentials at most quadratic.
• Correct mean motion does not prevent wave-packet spreading or splitting.
• Ehrenfest’s theorem does not by itself derive definite trajectories, decoherence, or the full classical limit.
• A broad or multimodal state can make its mean a poor description of any likely measurement result.

Canonical Treatment

The derivation, closure conditions, finite-width expansion, moment hierarchy, examples, exercises, and references are at Ehrenfest Theorem.

Related Lookup

• Heisenberg Equation
• Ehrenfest Theorem Revisited
• Correspondence Principle