Ehrenfest Theorem

Statement

For a Hamiltonian

$$H = \frac{p^2}{2m}+V(x),$$

Ehrenfest theorem gives

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

Equivalently,

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

General Form

For an operator $A(t)$ with possible explicit time dependence,

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

when the domains and differentiability assumptions are satisfied.

Assumptions

• The system evolves under closed-system Schrödinger dynamics.
• $x$, $p$, $H$, and the commutators are meaningful on the states considered.
• Boundary terms vanish when differential-operator manipulations are used.
• Replacing $\langle V'(x)\rangle$ by $V'(\langle x\rangle)$ requires an additional approximation.

Quantum-Mechanical Meaning

The theorem is exact for the stated Hamiltonian. It becomes a classical equation for the mean only when

$$\langle V'(x)\rangle \approx V'(\langle x\rangle).$$

This equality is exact for potentials at most quadratic in $x$ and approximate for sufficiently narrow wave packets in slowly varying forces.

Canonical Links

• Ehrenfest Theorem
• Ehrenfest Theorem Overview
• Ehrenfest Theorem Revisited
• Heisenberg Equation
• Conservation Laws
• Gaussian Wave Packets

Common Mistakes

• Saying the theorem proves that particles follow classical trajectories.
• Replacing $\langle V'(x)\rangle$ by $V'(\langle x\rangle)$ without checking localization.
• Ignoring packet spreading, splitting, or interference.
• Treating expectation values as a complete description of a quantum state.
• Forgetting domains and boundary conditions for unbounded operators.

Quick Check

For a harmonic oscillator potential $V(x)=m\omega^2x^2/2$, why does the mean position obey the exact classical oscillator equation?

Solution

Here $V'(x)=m\omega^2x$, so $\langle V'(x)\rangle=m\omega^2\langle x\rangle$. Ehrenfest theorem gives

$$\frac{d^2}{dt^2}\langle x\rangle+\omega^2\langle x\rangle=0.$$

References

• P. Ehrenfest, “Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik,” Zeitschrift für Physik 45, 455-457, 1927.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.