Ehrenfest's Theorem

Quantum mechanics replaced trajectories with wavefunctions, forces with potentials acting on phases, and certainty with statistics. So where did Newton go? He cannot simply be wrong: baseballs, planets, and dust grains obey $F = ma$ to every decimal anyone has checked, and they are made of quantum particles. The bridge — stated and proved by Paul Ehrenfest in 1927, two pages that every textbook has retold since — is that Newton survives on average. The expectation values of position and momentum, the ensemble-mean quantities built in Expectation Values, obey

\frac{d\langle x\rangle}{dt} = \frac{\langle p\rangle}{m}, \qquad \frac{d\langle p\rangle}{dt} = -\left\langle\frac{\partial V}{\partial x}\right\rangle,

which look exactly like Hamilton's classical equations with averaging brackets bolted on. The first relation was earned in Momentum Operator — it is what licensed the operator's name. This article derives the second, assembles the pair into "Newton for averages," and then does the thing that separates understanding from folklore: it reads the fine print. The average force $\langle -V'(x)\rangle$ is not the force at the average position $-V'(\langle x\rangle)$ , and the daylight between those two expressions is precisely where classical mechanics ends. For potentials no steeper than quadratic the two coincide and quantum centroids move on exact classical trajectories — forever, as a worked oscillator example shows. For real anharmonic potentials the packet's width feeds back into its own motion, and a second worked example puts hard numbers on the failure. Ehrenfest's theorem is the correspondence principle with the vagueness removed: not "quantum becomes classical somehow," but an exact identity plus a controlled approximation whose error term you can compute.

Key takeaways

Ehrenfest pair: $d\langle x\rangle/dt = \langle p\rangle/m$ (owned by Momentum Operator) and $d\langle p\rangle/dt = -\langle\partial V/\partial x\rangle$ (derived here).
Combined: $m\,d^2\langle x\rangle/dt^2=-\langle V'(x)\rangle$ ---an exact equation for the ensemble-mean position, which becomes Newton's second law only when $\langle V'(x)\rangle=V'(\langle x\rangle)$ .
The catch: $\langle V'(x)\rangle \neq V'(\langle x\rangle)$ in general — the mean force is not the force at the mean.
Taylor bookkeeping: $\langle V'(x)\rangle = V'(\langle x\rangle) + \tfrac12\sigma_x^2V'''(\langle x\rangle) + \cdots$ — the error is packet width times potential curvature-of-curvature.
Exactly classical cases: $V$ constant, linear, or quadratic ( $V''' = 0$ ): free particles, uniform fields, and harmonic oscillators have centroids on exact classical trajectories.
Classicality condition: packet narrow compared with the scale over which $V'$ varies — why baseballs are Newtonian and electrons in molecules are not.
Key idea: quantum expectation values obey classical-looking equations of motion exactly; genuine Newtonian dynamics for the centroid emerges only when the mean force equals the force at the mean.

The core insight

Expectation values of position and momentum obey $d\langle x\rangle/dt = \langle p\rangle/m$ and $d\langle p\rangle/dt = -\langle\partial V/\partial x\rangle$ — classical-like equations of motion that become genuinely Newtonian when the packet is narrow enough that the mean force equals the force at the mean position.

Where this lives today

"Ehrenfest dynamics" is a standard method in computational chemistry and materials science: the electrons are propagated quantum mechanically while the nuclei move classically under the mean electronic force — the same force law expressed by equation (2), used as a practical mixed quantum–classical approximation. It powers simulations of photochemistry, ion bombardment, and electron–phonon energy transfer, and its known failure modes are this article's fine print institutionalized: whenever the electronic packet splits across distinct outcomes (a molecule deciding between dissociation channels), the mean force steers the nuclei down a nonexistent average path, and practitioners switch to "surface hopping" schemes that restore the split. Method selection in a 2026 nonadiabatic-dynamics code is literally an audit of when brackets may slide inside primes. At the other extreme, the exactly classical harmonic-oscillator case is useful in precision quantum control. Trapped-ion systems use highly harmonic motional potentials, and the Ehrenfest equations describe the motion of the wave-packet centroid exactly for a quadratic potential. This classical-looking motion underlies the interpretation of phase-space displacements and related motional-state control techniques.

The physical picture

Run a quantum movie of a wavepacket thrown across a potential landscape. The packet blurs, breathes, and develops ripples — nothing about it looks like a marble. Now overlay a single dot at the packet's centroid $\langle x\rangle(t)$ , the balance point of the probability mist. Ehrenfest's theorem says that the packet's centroid moves according to a classical-like equation: its acceleration times the mass equals $-\langle V'(x)\rangle$ at every instant, exactly, for any quantum state. The theorem is not approximate. What is approximate — sometimes brilliantly, sometimes catastrophically — is replacing the packet-averaged force by the force at the dot.

Figure

The centroid rides a classical trajectory while the packet blurs around it

Where can this go wrong, if the theorem is exact? In the meaning of the force. The packet does not sample the potential's slope at one point; it samples the slope everywhere it has amplitude, weighted by $|\Psi|^2$ . If the slope varies across the packet — if $V'$ curves on the scale of the packet's width — the average slope can differ badly from the slope at the centre. A packet straddling a region where the potential varies appreciably samples different parts of the landscape at once; a packet split by a barrier feels two landscapes at once, and its centroid then moves like nothing in Newton's world.

Figure

Mean force versus force at the mean — where Newton leaks away

Hold the two panels as the theorem's constitution: the dot always obeys a force law (exact); it obeys Newton's force law when the packet is narrow against the potential's wiggles (approximate, with a computable error).

The theorem, completed

The kinematic half is already on the shelf: Momentum Operator's third derivation card proved $d\langle x\rangle/dt = \langle p\rangle/m$ — the centroid's velocity is the mean momentum over the mass — and that result is recalled here, not re-derived. What remains is the dynamic half: how the mean momentum itself changes.

Derivation ANewton's second law for averages

We differentiate $\langle p\rangle$ under the integral, let the Schrödinger equation replace every time derivative, and watch a beautiful division of labour: the kinetic terms annihilate each other completely (free flight does not change momentum), while the potential terms — which cancelled in the norm and current derivations — here survive precisely in the combination $V\partial_x - \partial_x V$ , whose whole content is the multiplication-rule commutator $-V'$ . The boundary-term hygiene is the same as in Conservation of Normalization; the payoff is the force law.

Step 1. Set up the derivative. $\dfrac{d\langle p\rangle}{dt} = \dfrac{d}{dt}\displaystyle\int\Psi^*(-i\hbar\,\partial_x)\Psi\,dx = -i\hbar\displaystyle\int\left(\partial_t\Psi^*\,\partial_x\Psi + \Psi^*\,\partial_x\partial_t\Psi\right)dx$ . Why: fixed limits let the derivative inside; the product rule splits it across the two state factors, with $\partial_t$ and $\partial_x$ commuting on smooth solutions.

Step 2. Insert the equation of motion twice. With $\partial_t\Psi = \tfrac{i\hbar}{2m}\partial_x^2\Psi - \tfrac{i}{\hbar}V\Psi$ and its conjugate ( $V$ real), the integrand splits into kinetic×kinetic-gradient terms and potential terms. Why: the state has no dynamics of its own — the one law supplies every time derivative, exactly as in the current and norm derivations of this chapter.

Step 3. Kill the kinetic block. The kinetic contribution is $-\tfrac{\hbar^2}{2m}\!\int\!\left(\partial_x^2\Psi^*\,\partial_x\Psi - \Psi^*\,\partial_x^3\Psi\right)dx$ ; integrate by parts (twice on the second term) and the terms cancel, with the boundary terms vanishing for sufficiently well-behaved normalizable states. Why: physically, a free particle's $\langle p\rangle$ is constant — momentum conservation for $V = 0$ — so the kinetic sector must give zero.

Step 4. Collect the potential block. What survives is $-i\hbar\cdot\left(-\tfrac{i}{\hbar}\right)\!\int\!\Psi^*\left[V\,\partial_x\Psi - \partial_x(V\Psi)\right]dx = -\int\Psi^*\,V'(x)\,\Psi\,dx$ , by the product rule $\partial_x(V\Psi) = V'\Psi + V\partial_x\Psi$ . Hence

\frac{d\langle p\rangle}{dt} = -\left\langle\frac{\partial V}{\partial x}\right\rangle \qquad (1)

Why: the mean momentum changes at the rate set by the packet-averaged force. Nothing was assumed about the state: (1) is an exact identity for every normalizable solution.

Step 5. Assemble the pair. Differentiating the kinematic half once more and inserting (1):

m\,\frac{d^2\langle x\rangle}{dt^2} = -\left\langle V'(x)\right\rangle \qquad (2)

Why: the centroid obeys mass-times-acceleration-equals-mean-force — Newton's second law with ensemble brackets. The remaining question, when may the brackets slide inside the prime, is the next card's business. (The same theorem in its general costume — $d\langle Q\rangle/dt$ for any observable, via the commutator with $\hat H$ — is the deeper-register sibling article Ehrenfest's Theorem of the Quantum Dynamics chapter; this article owns the position–momentum pair.)

Go deeper · Derivation B · 1 minWhen the brackets slide inside — the classicality audit

Derivation BWhen the brackets slide inside — the classicality audit

Equation (2) is exact but not yet Newton: Newton needs the force at the centroid, $-V'(\langle x\rangle)$ , and (2) delivers the averaged force $-\langle V'(x)\rangle$ . This card computes the difference once and for all, by Taylor-expanding the force across the packet. The result is a precise statement of the correspondence principle: it gives an explicit leading correction, proportional to the packet's variance and the potential's third derivative, that vanishes for quadratic potentials and lets you estimate when the quantum motion is well approximated by classical mechanics.

Step 1. Expand the force about the centroid. With $\delta = x - \langle x\rangle$ : $V'(x) = V'(\langle x\rangle) + \delta\,V''(\langle x\rangle) + \tfrac{\delta^2}{2}V'''(\langle x\rangle) + \cdots$ . Why: the packet samples the force in a neighbourhood; Taylor organizes the neighbourhood by moments of $\delta$ .

Step 2. Average term by term. $\langle\delta\rangle = 0$ by definition of the centroid, and $\langle\delta^2\rangle = \sigma_x^2$ (the variance, whose machinery is the next article's, used here in preview):

\langle V'(x)\rangle = V'(\langle x\rangle) + \frac{\sigma_x^2}{2}\,V'''(\langle x\rangle) + \cdots \qquad (3)

Why: the linear term dies exactly — this is why the theorem is so forgiving — and the first surviving correction couples the packet's width to the third derivative of the potential: the force must CURVE across the packet before quantum spread back-reacts on the mean motion.

Step 3. Read off the exact cases. If $V''' \equiv 0$ — $V$ constant, linear, or quadratic — every correction term vanishes and $m\,d^2\langle x\rangle/dt^2 = -V'(\langle x\rangle)$ exactly: free particles, uniform fields (gravity, capacitor plates), and harmonic oscillators have centroids on rigorously classical trajectories, whatever the state, however wide the packet, forever. Why: three derivatives of a quadratic are zero; the harmonic oscillator's special citizenship in quantum mechanics — coherent states, classical limits — starts at this fact.

Step 4. State the classicality criterion. Newton's law holds when the packet is narrow enough that the force does not vary appreciably across it, so that $\langle V'(x)\rangle\approx V'(\langle x\rangle)$ . For a smooth potential, the leading correction is $\frac{\sigma_x^2}{2}V'''(\langle x\rangle)$ , so away from points where $V'(\langle x\rangle)=0$ its relative size is roughly $\frac{\sigma_x^2}{2}|V'''/V'|$ . A narrow packet in a slowly varying potential therefore follows a classical trajectory, while a broad packet or strongly varying potential can produce significant quantum corrections.

Limiting cases

Case	Result	Meaning
$V = 0$	$\langle p\rangle$ constant, $\langle x\rangle$ linear in $t$	momentum conservation for averages; the packet still spreads
$V = -Fx$ (uniform force)	exact classical parabola for $\langle x\rangle(t)$	linear potential: $V''' = 0$ , Newton exact
$V = \tfrac12 m\omega^2x^2$	$\langle x\rangle(t) = A\cos(\omega t + \phi)$ exactly, any state	the oscillator's centroid is rigorously classical — worked example 1
narrow packet, smooth $V$	Newton to relative error $\tfrac{\sigma_x^2}{2}\lvert V'''/V'\rvert$	the classical world's licence, with its error bar
packet split by a barrier	centroid can sit where no particle ever is; no classical reading	Ehrenfest exact, Newton meaningless — brackets refuse to slide inside

Common trap“Ehrenfest's theorem says quantum systems obey classical mechanics on average”

Half true, and the false half costs real marks. The exact statement (2) equates the centroid's acceleration to the mean force — an ensemble average over the packet — not to the classical force at the centroid. The two coincide exactly for potentials that are at most quadratic, and approximately for sufficiently narrow packets when the force varies little across the packet. Concrete failure: a Gaussian packet incident on a barrier splits into transmitted and reflected pieces; the centroid ends up between them, gliding through a region where the particle is almost never found, on a "trajectory" no classical particle in that potential could follow. Nothing broke — equation (2) holds to the last decimal — but the mean of two classical motions is not a classical motion. A second, subtler misreading: the theorem directly governs only the first moments, $\langle x\rangle$ and $\langle p\rangle$ . Packet spreading, tunnelling probabilities, interference, and higher moments require additional information about the wavefunction. Thus “the wavepacket moves classically” is never a licence to think classically about what the packet is doing around its own centroid.

Worked examples

1 · The oscillator's exactly classical centroid (confidence builder)

An electron sits in a harmonic trap of frequency $\omega = 1.0\times10^{15}\ \mathrm{rad/s}$ (a molecular-vibration scale), prepared in the equal superposition $\Psi = \frac{1}{\sqrt2}(\psi_0 e^{-iE_0t/\hbar} + \psi_1 e^{-iE_1t/\hbar})$ of the two lowest oscillator states (from Harmonic Oscillator, Analytic Method). Using the cross matrix element $\langle\psi_0|x|\psi_1\rangle = \sqrt{\hbar/2m\omega}$ , find $\langle x\rangle(t)$ and verify it solves Newton's equation for this potential exactly.

Step 1. By the beat machinery of Expectation Values (equation (4) there), with $\langle x\rangle_0 = \langle x\rangle_1 = 0$ by symmetry: $\langle x\rangle(t) = \sqrt{\dfrac{\hbar}{2m\omega}}\cos(\omega_{10}t)$ , where $\omega_{10} = (E_1 - E_0)/\hbar = \omega$ — for the oscillator, the Bohr frequency is the classical frequency, because the ladder is evenly spaced.

Step 2. Amplitude: $\sqrt{\dfrac{\hbar}{2m\omega}} = \sqrt{\dfrac{1.055\times10^{-34}}{2(9.109\times10^{-31})(10^{15})}} = \boxed{0.24\ \mathrm{nm}}$ — the quantum length scale of the trap.

Step 3. Newton audit: classical motion in $V = \tfrac12m\omega^2x^2$ obeys $\ddot x = -\omega^2x$ ; differentiate Step 1 twice: $\dfrac{d^2\langle x\rangle}{dt^2} = -\omega^2\langle x\rangle$ ✓ — exactly, at every instant, with no narrow-packet caveat. $V''' = 0$ has done its work.

Step 4. Read the period: $T = 2\pi/\omega = 6.3\ \mathrm{fs}$ . An ensemble-mean position measurement on this manifestly quantum state — two discrete levels, nothing classical about it — traces a pure classical cosine at $0.24\ \mathrm{nm}$ amplitude. This exactness is (the packets whose expectation values show classical-looking motion while also preserving their wave-packet shape) — a story owned by the ONED chapter's later articles.

2 · The classicality audit for a quartic trap (exam level)

A packet of width $\sigma_x$ sits centred at $x_0$ in the anharmonic potential $V = \lambda x^4$ . (a) Compute the exact mean force for a Gaussian packet and the naive force at the centroid. (b) Evaluate the relative error for $\sigma_x = x_0/2$ , and for a dust-grain case $\sigma_x = 10^{-9}x_0$ .

Step 1. The force law: $V' = 4\lambda x^3$ , so $\langle V'\rangle = 4\lambda\langle x^3\rangle$ . For a Gaussian centred at $x_0$ with variance $\sigma_x^2$ , the third moment is $\langle x^3\rangle = x_0^3 + 3x_0\sigma_x^2$ (odd central moments vanish). Hence $\langle V'\rangle = 4\lambda x_0^3\left(1 + 3\sigma_x^2/x_0^2\right)$ .

Step 2. The naive force: $V'(\langle x\rangle) = 4\lambda x_0^3$ . Relative error: $\boxed{3\sigma_x^2/x_0^2}$ — which is exactly the Taylor prediction (3): $\tfrac{\sigma_x^2}{2}V''' / V' = \tfrac{\sigma_x^2}{2}\cdot 24\lambda x_0/(4\lambda x_0^3) = 3\sigma_x^2/x_0^2$ ✓, higher Gaussian corrections vanishing because $V''''$ 's term involves $\langle\delta^3\rangle = 0$ .

Step 3. Quantum-scale packet, $\sigma_x = x_0/2$ : error $= 3/4 = \boxed{75\%}$ — the centroid's acceleration is three-quarters again what Newton predicts; a trajectory calculation is quantitatively worthless, and genuinely wave-mechanical tools are mandatory.

Step 4. Dust grain, $\sigma_x = 10^{-9}x_0$ : error $= 3\times10^{-18}$ — eighteen digits of Newtonian fidelity. Same theorem, same formula; the only difference is the ratio of packet width to potential scale. This pair of numbers is the quantum-classical boundary, located not by philosophy but by arithmetic.

Origins · 1 minPaul Ehrenfest and how this was discovered

Paul Ehrenfest

1880–1933 · Vienna, St. Petersburg, Leiden

Einstein called him "the best teacher in our profession" — and physics used Ehrenfest above all as its conscience: the man who demanded that revolutionary formalisms explain themselves in plain physical terms. A Boltzmann student from Vienna who succeeded Lorentz at Leiden in 1912, he had a genius for the clarifying question and the clean minimal argument — his adiabatic principle guided the old quantum theory, his classification of phase transitions still names them, and his 1927 two-page note proved the theorem of this article: the new wave mechanics, asked what became of $F = ma$ , answers with expectation values. It was quintessential Ehrenfest — a bridge, built in an afternoon, between the mechanics everyone trusted and the mechanics no one yet understood, with the honest fine print visible from the start. His seminars in Leiden, and the legendary arguments he hosted and refereed between Bohr and Einstein, shaped how a generation reasoned about quanta. The end of his story is dark: exhausted, depressed, and despairing — of his powers against the accelerating new physics, and over his youngest son's Down syndrome — he took his own life and his son's in September 1933. Physics lost its conscience; the theorem keeps his name on the exact spot where the classical world begins.

History

1913

— Ehrenfest's adiabatic hypothesis becomes a guiding principle of the old quantum theory.

1923

— The correspondence principle demands classical mechanics emerge from the quantum theory in the right limit.

1927

— Ehrenfest's two-page paper: expectation values of the new wave mechanics obey Newton-like equations.

1930

— Textbook treatments locate the fine print: mean force versus force at the mean, and the quadratic-potential exactness.

1952

— Bohm's pilot-wave theory offers a different classical bridge, with trajectories guided by the same expectation-value structure.

Going further

Texts. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §1.5 & §3.5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 6 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.2 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Ehrenfest, Z. Phys. 45, 455 (1927). The key results in this article are: (1) the $d\langle p\rangle/dt$ derivation — kinetic cancellation by double integration by parts and the surviving $-\langle V'\rangle$ ; (2) the Taylor audit $\langle V'\rangle = V'(\langle x\rangle) + \tfrac{\sigma_x^2}{2}V'''(\langle x\rangle) + \cdots$ ; (3) worked example 1 — QHO cross element $\langle 0|x|1\rangle = \sqrt{\hbar/2m\omega}$ by direct integration, amplitude $0.24\ \mathrm{nm}$ at $\omega = 10^{15}\ \mathrm{rad/s}$ , $\ddot{\langle x\rangle} = -\omega^2\langle x\rangle$ exact; (4) worked example 2 — Gaussian third moment $x_0^3 + 3x_0\sigma_x^2$ , relative errors $75\%$ ( $\sigma_x = x_0/2$ ) and $3\times10^{-18}$ ( $\sigma_x = 10^{-9}x_0$ ). The kinematic half $d\langle x\rangle/dt = \langle p\rangle/m$ is owned by Momentum Operator; the general commutator form $d\langle Q\rangle/dt = \tfrac{i}{\hbar}\langle[\hat H,\hat Q]\rangle + \langle\partial_t\hat Q\rangle$ is owned by the Quantum Dynamics chapter's Ehrenfest's Theorem (the deeper-register sibling of this article).

Prerequisites. Time-Dependent Schrödinger Equation · The Wavefunction · Born Statistical Interpretation · Probability Density and Normalization · Momentum Operator · Expectation Values. Leads to. Standard Deviation and Quantum Uncertainty · Breakdown of Classical Physics (the loop closed) · Stationary States and Separation of Variables · the Quantum Dynamics chapter's general Ehrenfest machinery.

Why are there two articles called Ehrenfest's Theorem in this track?

Deliberate design, at two registers. This article owns the position–momentum pair in wavefunction language — integrals, integration by parts, the Taylor audit — which is where the physics of the classical limit lives. The Quantum Dynamics chapter's sibling derives the general equation of motion $d\langle Q\rangle/dt = \tfrac{i}{\hbar}\langle[\hat H, \hat Q]\rangle + \langle\partial_t\hat Q\rangle$ for any observable, in commutator language, and re-obtains this article's pair as two lines of algebra ( $[\hat H, \hat x]$ and $[\hat H, \hat p]$ ). Same theorem, deeper machinery: the commutator form exposes conservation laws instantly ( $[\hat H, \hat Q] = 0 \Rightarrow \langle Q\rangle$ frozen) and generalizes to spin and beyond, where "integrate by parts" means nothing. Read this one for the meaning, that one for the engine.

If the centroid of a barrier-split packet moves unclassically, has momentum conservation failed?

No — audit which theorem says what. Equation (1) with the actual potential holds exactly throughout the split: while the packet overlaps the barrier, the barrier exerts a genuine mean force, and after the split the centroid coasts at constant $\langle p\rangle$ — a perfectly lawful average of the transmitted and reflected momenta. What fails is only the interpretation of the centroid as "where a particle is going": no single detection event follows the centroid's path. Ensemble language, as always in this chapter, dissolves the paradox: Ehrenfest governs the mean of many one-shot experiments, and means of bimodal distributions sit in their empty middles — the same lesson as the mean-at-a-node example of Position Operator, now in motion.

Does Ehrenfest's theorem PROVE the classical limit of quantum mechanics?

It proves the cleanest piece and honestly labels the rest. What it delivers: exact classical centroid motion for $V''' = 0$ , and controlled-error classical motion for narrow packets in smooth potentials. What it does not deliver: (a) why packets stay narrow — free spreading (see Free Particle and Wave Packets) eventually violates the narrowness condition, and for chaotic classical dynamics the violation arrives exponentially fast; (b) why macroscopic objects never show up in superpositions of distinct trajectories in the first place — that is decoherence, the environment's continuous measurement of position, owned by the Entanglement chapter. The modern consensus: Ehrenfest + decoherence + environment-induced localization jointly explain classicality; Ehrenfest alone is the skeleton.

Is there an Ehrenfest theorem for energy — does ⟨H⟩ obey a classical equation too?

Yes, and it is the simplest of all: for a time-independent Hamiltonian, $d\langle H\rangle/dt = 0$ exactly — mean energy is conserved, as Expectation Values' first worked example exploited. (The general commutator form makes this one line: $[\hat H, \hat H] = 0$ .) With a time-dependent Hamiltonian, $d\langle H\rangle/dt = \langle\partial_t\hat H\rangle$ — energy changes at the rate the Hamiltonian is externally cranked, the quantum version of the classical work-energy bookkeeping. Between them, these give the average-level laws of thermodynamic work on quantum systems, a thread the track's later chapters pick up.

The oscillator centroid is exactly classical — so is an oscillator 'classical'?

Its first moment is classical; the state is as quantum as anything in the theory. The two-level superposition of worked example 1 has discrete energies, zero-point motion, and interference structure that no classical oscillator possesses; measure anything beyond $\langle x\rangle$ — the energy distribution, the width $\sigma_x(t)$ , correlations — and the quantum character is immediate. The sharpest illustration: the oscillator's ground state has an exactly classical centroid too ( $\langle x\rangle = 0$ , forever, which classically means a particle at rest at the bottom) while being irreducibly fuzzy, with $\sigma_x\sigma_p = \hbar/2$ exactly. Ehrenfest exactness is a statement about one moment of the distribution, never about the physics as a whole — the coherent states of the ONED chapter are the closest an oscillator state comes to being classical in all low moments at once.

How does the σₓ²V‴ correction connect to actual ℏ? It looks ℏ-free.

Sharp eye — equation (3) is a statement about any probability distribution and contains no $\hbar$ explicitly. Quantum mechanics enters through the floor on $\sigma_x$ : a classical ensemble could in principle have $\sigma_x = 0$ (all systems at exactly $x_0$ ) and Newton would be exact for any potential; quantum states cannot, because squeezing $\sigma_x$ inflates $\sigma_p$ (the uncertainty structure of the next article) and the kinetic energy with it, so the dynamically achievable $\sigma_x$ has an $\hbar$ -set minimum for given energy. Track the argument through and the honest small parameter of the classical limit is $\hbar$ against the classical action scale of the problem — the same dimensionless ratio that controls the WKB approximation in the track's later chapters. Ehrenfest's correction term is where that ratio first becomes visible in this course.

Do Ehrenfest trajectories relate to Bohmian trajectories?

They are different objects that agree only in special cases — a comparison worth one paragraph because both get called "quantum trajectories." The Ehrenfest centroid is a single ensemble-average path; Bohmian mechanics (flagged in Probability Current and the Continuity Equation) assigns each individual particle a definite trajectory riding the flow field $v = j/\rho$ , with the ensemble of Bohmian paths distributed as $|\Psi|^2$ . The mean of the Bohmian ensemble reproduces $\langle x\rangle(t)$ exactly — both frameworks respect the same moments — but individual Bohmian paths can do things the centroid never does (swerve through interference fringes, refuse to cross). At the barrier split: the centroid glides through the middle; each Bohmian path goes definitely one way or the other. Neither is "the" trajectory of standard quantum mechanics, which declines to supply one; they are two disciplined ways of drawing lines under a distribution-valued theory.