# Ehrenfest theorem

The **Ehrenfest theorem** is a result of quantum mechanics that relates the time derivative of the expectation value of an observable to the expectation value of its commutator with the Hamiltonian (the operator representing a system's total energy), and it shows that expectation values of position and momentum obey equations resembling [Newton's laws of motion](https://www.edgepedia.org/newtons-laws-of-motion).<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup><sup> • </sup><sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup> It was published by [Paul Ehrenfest](https://www.edgepedia.org/paul-ehrenfest) in 1927 and is historically the first result showing that classical physics can emerge from quantum physics as a kind of approximation.<sup>[3](https://arxiv.org/abs/1907.02354)</sup>

| Key fact | Detail |
|---|---|
| General statement | \( \frac{d}{dt}\langle A\rangle = \left\langle \frac{\partial A}{\partial t} \right\rangle + \frac{i}{\hbar}\langle[H,A]\rangle \) for an operator \( A \) in state \( \psi \)<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup> |
| Time-independent operator | \( \frac{d\langle A\rangle}{dt} = \frac{i}{\hbar}\langle[\hat H,A]\rangle \), with the commutator \([A,B] := AB-BA\)<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup> |
| Position and momentum | \( \frac{d\langle x\rangle}{dt} = \frac{\langle p\rangle}{m} \) and \( \frac{d\langle p\rangle}{dt} = -\left\langle \frac{dV}{dx} \right\rangle \)<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup> |
| Newton analogy | \( m\,\frac{d^2\langle x\rangle}{dt^2} = -\langle \nabla V \rangle \), the quantum mechanical equivalent of Newton's second law, with no dependence on \( \hbar \)<sup>[4](https://farside.ph.utexas.edu/teaching/389/lectures/node33.html)</sup> |
| Conserved quantities | Expectation values of operators that commute with the Hamiltonian are constants of the motion<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup> |
| Classical correspondence | Commutators are obtained from Poisson brackets by \( \{\cdot,\cdot\} \mapsto -\frac{i}{\hbar}[\cdot,\cdot] \); \( \{x,p\}=1 \) maps to \( [\hat x,\hat p]=i\hbar \)<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup> |
| Classical limit | Holds when the spatial extent \( \sigma_x \) of the wavefunction, typically of order the de Broglie wavelength, is much smaller than the variation length-scale of the potential<sup>[5](https://farside.ph.utexas.edu/teaching/qmech/Quantum/node36.html)</sup> |

## Statement and proof

For a Hermitian operator \( A \), such as position \( \hat x \), momentum \( \hat p \), or the Hamiltonian \( \hat H \), the expectation value in a wave function \( \psi(x,t) \) is \( \langle A \rangle := \langle \psi, A\psi \rangle \). Differentiating under the integral gives two terms from the time dependence of \( \psi \) and one from explicit time dependence of \( A \) itself.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup> Replacing the partial time derivatives of the state with the Hamiltonian using Schrödinger's equation, \( i\hbar\,\partial_t\psi = \hat H\psi \), and using the Hermiticity of \( A \), the derivative reduces to<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

\[ \frac{d \langle A\rangle }{dt} = \frac{i}{\hbar} \langle [\hat H,A] \rangle, \qquad [A,B] := AB-BA. \]

If \( A \) depends explicitly on time, the general form adds the term \( \langle \partial A/\partial t \rangle \).<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup> The same relation follows from evaluating the time derivative of \( \langle\Psi(t)|\hat A|\Psi(t)\rangle \) with the time-evolution operator \( \hat U(t,0) \).<sup>[6](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S19.html)</sup>

An immediate consequence concerns conservation. An operator that commutes with the Hamiltonian has a constant expectation value, so such operators define conserved quantities, the quantum analog of the classical condition \( \{H,A\}=0 \) for a quantity to be constant in time. The Hamiltonian itself is always a conserved quantity in this sense.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)</sup><sup> • </sup><sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

## The classical equations of motion

Applying the theorem to position and momentum for a Hamiltonian \( H = p^2/2m + V(x) \) gives<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

\[ \frac{d \langle x\rangle }{dt} = \frac{\langle p\rangle}{m}, \qquad \frac{d\langle p\rangle }{dt} = -\left\langle \frac{dV}{d x} \right\rangle, \qquad m \frac{d^2\langle x\rangle}{d t^2} = -\left\langle \frac{\partial V}{\partial x} \right\rangle. \]

In vector form, \( m\,\frac{d^2\langle \mathbf{x}\rangle}{dt^2} = -\langle \nabla V(\mathbf{x})\rangle \). This equation contains no \( \hbar \), but it does not guarantee that the center of a wavepacket follows an exact classical trajectory, since its acceleration depends on the average force, which generally differs from the force at the center's average position.<sup>[4](https://farside.ph.utexas.edu/teaching/389/lectures/node33.html)</sup><sup> • </sup><sup>[11](https://mathworld.wolfram.com/EhrenfestsTheorem.html)</sup> Because it is written in terms of expectation values, the result is independent of whether the Heisenberg or [Schrödinger picture](https://www.edgepedia.org/schrodinger-picture) is used; the corresponding operator equation holds only for Heisenberg dynamical variables.<sup>[4](https://farside.ph.utexas.edu/teaching/389/lectures/node33.html)</sup>

The theorem also connects directly to the [Heisenberg picture](https://www.edgepedia.org/heisenberg-picture), in which operators carry the time evolution and obey \( \frac{d}{dt}\hat A = \frac{i}{\hbar}[\hat H,\hat A] \). In some cases the resulting equations are formally identical to the classical equations of motion; in general, however, the commutators generate new expressions, giving rise to an infinite hierarchy of equations.<sup>[6](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S19.html)</sup>

**Free particle.** For a [Gaussian wave packet](https://www.edgepedia.org/gaussian-wave-packet) with momentum \( p_0 \), the theorem gives \( \langle x \rangle = x_0 + \frac{p_0}{m}t \) and \( \langle p \rangle = p_0 \): the center of the packet moves with uniform velocity \( \langle p \rangle/m \), solving the classical equations of motion with \( V(x)=0 \), without a long explicit computation of the integrals.<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

## The classical limit and its conditions

The theorem is easily misinterpreted as a proof that classical mechanics falls out of quantum theory for free.<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup><sup> • </sup><sup>[7](https://www.physics2u.com/vault/ehrenfest-theorem.html)</sup> The second equation equates \( d\langle p\rangle/dt \) with the average force, not the force at the average position. Expanding \( V(x) \) about \( \langle x \rangle \), the higher-order terms are suppressed when the spatial extent \( \sigma_x \) of the wavefunction is narrow compared with the variation length-scale of the potential; in that case the equations follow the classical ones approximately, and reduce to them exactly when the force is an affine function of position, with \( \langle x \rangle \) playing the role of the particle displacement, and a measurement of \( x \) is almost certain to yield a result close to \( \langle x \rangle \).<sup>[5](https://farside.ph.utexas.edu/teaching/qmech/Quantum/node36.html)</sup><sup> • </sup><sup>[11](https://mathworld.wolfram.com/EhrenfestsTheorem.html)</sup> The spatial extent of a wavefunction is typically of order the de Broglie wavelength, so quantum mechanics corresponds to classical mechanics in the limit that this extent is negligible.<sup>[5](https://farside.ph.utexas.edu/teaching/qmech/Quantum/node36.html)</sup>

A further qualification is that the expectation value must be physically meaningful. If the state is a superposition of macroscopically distinct outcomes, the equation still holds but the expectation value does not describe the behavior of any individual outcome.<sup>[7](https://www.physics2u.com/vault/ehrenfest-theorem.html)</sup>

## Subtleties: the particle in a box

The infinite potential well exposes where the naive statement fails. For a particle in a box described by the sum of the two lowest-energy eigenstates, the position expectation value oscillates as \( \langle x \rangle = \frac{L}{2} - A\cos(\omega t) \) with frequency \( \omega = \frac{E_2-E_1}{\hbar} \) and amplitude \( A < \frac{L}{2} \). This is not a solution of the classical equation of motion, in which a particle feels no force except at the walls and bounces with uniform velocity.<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

The momentum relation \( \langle p \rangle = m\,\frac{d\langle x\rangle}{dt} \) still holds, but computing \( \langle V'(x)\rangle \) is subtle because the potential jumps at \( x=0 \) and \( x=L \) and its derivative diverges there. A careful treatment shows that \( \frac{d\langle p\rangle}{dt} \) reduces to a boundary term \( -\frac{\hbar^2}{2m}\left. |\partial_x\psi|^2 \right|^L_0 \), so a generalized form of the theorem is required when boundary terms ignored in the standard derivation are non-vanishing.<sup>[2](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)</sup>

## Rigorous status and the Ehrenfest time

The textbook derivation treats \( H \) and \( A \) formally, but both are typically unbounded self-adjoint operators, so making the equation rigorous requires that the domain of \( A \) be invariant under the propagator \( e^{-itH} \), that \( \langle A \rangle_{\psi(t)} \) be differentiable in \( t \), and that the commutator expectation value be well defined. Gero Friesecke, a mathematician working on quantum many-body problems, and Mario Koppen showed in 2009 that the first condition is essentially sufficient for the others, and obtained a rigorous version of Ehrenfest's equations under assumptions that include atomic and molecular Hamiltonians with Coulomb interactions, a case they noted had been missing from the literature.<sup>[8](https://arxiv.org/abs/0907.1877)</sup>

The theorem also sets a time scale for quantum-classical correspondence. In classically chaotic systems, a Gaussian wave packet spreads quantum mechanically, and the breakdown of classical-like behavior is characterized by an Ehrenfest time; in one dimension it is set by the [Lyapunov exponent](https://www.edgepedia.org/lyapunov-exponent) \( \lambda \) and a typical classical action \( A \).<sup>[9](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.65.035208)</sup> Beyond this time, wave-packet evolution separates into classical-like motion, spreading, and revivals, requiring descriptions beyond the Ehrestfest-time regime such as semiclassical measures on the torus.<sup>[10](https://arxiv.org/abs/1607.07572)</sup>

## References

1. [Time Derivative of Expectation Values, UCSD Physics 130a lecture notes](https://quantummechanics.ucsd.edu/ph130a/130_notes/node189.html)
2. [Ehrenfest's Theorem, Kasper Peeters, Mathematical Physics lecture notes, Durham University](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/ehrenfest.html)
3. [Ehrenfest's Theorem Revisited, arXiv:1907.02354](https://arxiv.org/abs/1907.02354)
4. [Ehrenfest Theorem, Richard Fitzpatrick, UT Austin lecture notes](https://farside.ph.utexas.edu/teaching/389/lectures/node33.html)
5. [Ehrenfest's Theorem, Richard Fitzpatrick, Quantum Mechanics, UT Austin](https://farside.ph.utexas.edu/teaching/qmech/Quantum/node36.html)
6. [Ehrenfest theorem and Heisenberg picture, PHYS223 lecture notes, Lancaster University](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S19.html)
7. [Ehrenfest's theorem and the classical limit, physics2u](https://www.physics2u.com/vault/ehrenfest-theorem.html)
8. [Gero Friesecke and Mario Koppen, On the Ehrenfest theorem of quantum mechanics, arXiv:0907.1877](https://arxiv.org/abs/0907.1877)
9. [Ehrenfest times for classically chaotic systems, Physical Review E 65, 035208](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.65.035208)
10. [Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time, arXiv:1607.07572](https://arxiv.org/abs/1607.07572)
11. [Ehrenfest's Theorem -- from Wolfram MathWorld](https://mathworld.wolfram.com/EhrenfestsTheorem.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Quantum operators and observables (overview)*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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