The average that obeys Newton

A wavepacket in a harmonic well behaves impeccably.

Start a Gaussian off-centre and at rest, integrate the Schrödinger equation, and follow the expectation value of position. It oscillates sinusoidally at the classical frequency, with the classical amplitude, in phase with a classical particle released from the same place — not approximately, but to whatever accuracy the arithmetic is done.

That agreement is exact and it is a trap. It is exact for every state of a harmonic oscillator, including states nobody would call classical, and it is exact for reasons that do not survive changing the potential.

The theorem, which is true #

Ehrenfest’s result of 1927 is short. For a Hamiltonian $H = p^2/2m + V(x)$, the expectation values obey

$$\frac{\mathrm{d}\langle x\rangle}{\mathrm{d}t} = \frac{\langle p\rangle}{m}, \qquad \frac{\mathrm{d}\langle p\rangle}{\mathrm{d}t} = \left\langle -\frac{\mathrm{d}V}{\mathrm{d}x}\right\rangle.$$

Both follow from the general rule that the rate of change of any expectation value is the expectation of its commutator with the Hamiltonian — the same algebraic structure that turns a symmetry into a conserved quantity in the classical theory, and both hold for any state and any potential whatever. Nothing is approximated.

The first equation is uncontroversial and says the average position moves at the average velocity. The second is the one that gets misread, and the misreading is subtle enough to survive in textbooks: it says the average momentum changes at a rate equal to the average of the force over the state.

Newton’s second law says something different. It says the momentum changes at a rate equal to the force at the position. For a particle those are the same statement because a particle is at one place. For a state spread over a region they are not, and the difference is the whole content of this essay.

Where the two part company #

Expand the force about the mean position:

$$\left\langle -V'(x)\right\rangle = -V'(\langle x\rangle) - \tfrac{1}{2}V'''(\langle x\rangle)\,\sigma_x^2 - \cdots$$

The leading correction involves the third derivative of the potential and the variance of the state. Both have to be there: a wide state in a potential whose force is linear feels no correction, and a narrow state in any potential feels almost none.

For a harmonic potential $V = \tfrac12 m\omega^2x^2$ the third derivative is zero, and so is every higher one. The correction vanishes identically for every state, and the centroid obeys Newton’s law exactly. That is the whole explanation of the opening figure, and it explains why the agreement there is not evidence of anything.

Only an exact parabola has a linear restoring force, and it is the linearity rather than the shape that makes the quantum average behave classically. A potential that merely resembles a parabola — close to one near the bottom, curving away further out — has a force whose average over a spread-out state is not the force at the average position, and the two part company as soon as the state is wide enough to notice the difference.

Change the potential to a quartic and the third derivative is not zero anywhere.

The centroid and the classical orbit leave together and part within a period. By the end of the run they are moving with different amplitudes and different phases, and the gap is larger than the starting displacement. The theorem has not failed anywhere: $\mathrm{d}\langle p\rangle/\mathrm{d}t$ equals $\langle -V'\rangle$ throughout, and the run confirms it. What has failed is the identification of $\langle V'\rangle$ with $V'(\langle x\rangle)$.

The physical reason is easy to state. In a quartic well the force grows faster than linearly, so the parts of the state further out are pulled back harder than the centroid would be. The average force therefore exceeds the force at the average, and the centre is dragged inward faster than a classical particle at that position would be. A state that is spread out is being acted on by a range of forces, and the range does not average to the middle one.

The correction has a name and a size #

It is worth putting a number on the leading correction, because “the average of the force is not the force at the average” is a phrase and $\tfrac12 V'''\sigma^2$ is a quantity.

For the quartic well used here, $V = \tfrac14\lambda x^4$, the third derivative is $6\lambda x$ and the correction to the force is $3\lambda\langle x\rangle\sigma_x^2$. Compared with the classical force $\lambda\langle x\rangle^3$, the fractional error is $3\sigma_x^2/\langle x\rangle^2$ — three times the square of the ratio of the packet’s width to its displacement.

That has an unwelcome property. It grows as the centroid approaches the origin, where $\langle x\rangle$ is small and $\sigma_x$ is not, so the correspondence is worst exactly at the moment of the swing when the particle is moving fastest and the trajectory is most definite-looking. Averaged over a period the error accumulates rather than cancelling, which is why the two curves in the figure separate steadily instead of wobbling about one another.

For a general potential the same expression says which regions are dangerous: wherever the force has curvature comparable to itself over the width of the state. Near a turning point, near a barrier, near the centre of an anharmonic well — those are the places, and they are the interesting places.

So what does give classical mechanics #

The condition is not that the object be large, and it is not that Planck’s constant be small — that constant is a constant. The condition is that

$$\left|V'''(\langle x\rangle)\right|\sigma_x^2 \ll \left|V'(\langle x\rangle)\right|,$$

which says the state must be narrow compared with the distance over which the force changes appreciably.

For a football that is comfortably satisfied and by an enormous margin, because $\sigma_x$ can be $10^{-15}$ m and the potential varies over metres. For an electron in an atom it is not satisfied at all: the state is the size of the region over which the Coulomb force changes by order itself, so the correction is not a correction.

Two consequences of putting it this way are worth having. First, a macroscopic object in a sufficiently sharp potential can lose the correspondence — the criterion involves the potential as much as the mass. Second, the criterion is time-dependent, because $\sigma_x$ is.

A free packet spreads without limit, so any criterion phrased in terms of its width comes with a clock attached. That is the awkward part of the correspondence: it is not a property a state has, but a property it has for a while, and how long depends on the potential it is sitting in.

A packet spreads. In free space its width grows without limit; in a bound anharmonic potential it spreads and eventually wraps around, since different parts of it have different energies and therefore different frequencies. Whatever narrowness the correspondence needed at the start, it will not have later, and the question becomes how long the agreement lasts rather than whether it holds.

What gives classical mechanics, then, is not a limit in which the packet becomes a point. It is a regime in which the potential changes slowly enough across the packet’s actual width that the average of the force and the force at the average are indistinguishable to whatever precision is being demanded. That is a statement about a potential and a state together, not about either alone.

The reciprocity — which is not negotiable — makes it worse than it looks. Making $\sigma_x$ small requires a large $\sigma_p$, which makes the packet spread faster, so a state narrow enough to satisfy the criterion comfortably is a state that will violate it soon. There is an optimum and it is not a very good one.

Why “ħ → 0” is a bad way to say it #

The classical limit is often written as $\hbar\to 0$, which is a limit of a constant and therefore not a limit of anything.

What the phrase means, when it means something, is that some dimensionless ratio involving $\hbar$ is small — typically an action divided by $\hbar$, or a de Broglie wavelength divided by a length over which the potential changes. Those are properties of the situation, and the same system can have a large one in one regime and a small one in another. An electron in a magnetic field is classical along the field and quantised across it.

Stating the criterion as a ratio also makes clear why it cannot be met once and for all. The relevant length for a bound state changes as the state evolves, so a system can be well inside the classical regime at one moment and outside it later, with nothing having been done to it. That is the content of the timescales below, and it is invisible in a formulation that treats the classical limit as a property of a constant.

How long the correspondence lasts #

For a smooth bound potential the departure grows as a power of time, and the correspondence survives for many periods before the accumulated error matters. That is the ordinary case and it is why classical mechanics works for planets.

The clearest failure is a maximum. Put a state near the top of a hill and it splits, with the two halves going opposite ways — and the centroid then follows neither of them. It sits between the two packets, in a place where nothing is happening, describing a particle that is not there. Ehrenfest’s theorem is still exactly true; it has simply stopped being about anything.

Two situations break it much sooner.

A state at an unstable point. Place a packet at the top of a barrier and the two halves fall opposite ways. The centroid stays at the top, which is a correct expectation value and a description of nothing: the object is not at the top, it is on both sides. Every statement in this essay remains exactly true and the centroid has stopped being a useful summary of the state.

A chaotic system. Where neighbouring classical trajectories separate exponentially, so do the parts of a spread packet, and the width grows exponentially rather than as a power. The correspondence then lasts only until the exponential growth has taken the width from its initial value to the size of the system, which takes a time proportional to the logarithm of the ratio.

Chaos shortens the clock exponentially. In a system whose classical trajectories separate at an exponential rate, a quantum packet spreads at that same rate — so the time over which its centroid tracks a trajectory grows only as the logarithm of anything one might do to make the state narrower. That is why the correspondence is fragile in exactly the systems where a classical description would be most useful.

A logarithm is a brutal thing to have in a lifetime. Improving the initial state by ten orders of magnitude buys a factor of $\ln 10^{10}$, which is about twenty-three Lyapunov times, and no amount of care buys more than that arithmetic allows. That is the Ehrenfest time, and for macroscopic chaotic systems it is embarrassingly short: the standard estimate for one of Saturn’s tumbling moons is a few decades, after which an isolated Hyperion should be in a superposition of orientations.

It is not, and the reason is not in this essay: it is decoherence, which continually re-narrows the state by entangling it with everything around it. What Ehrenfest’s theorem cannot supply, and what the classical limit actually requires, is an environment.

The theorem’s real job #

None of the above is a criticism of Ehrenfest’s theorem, which does something valuable and is simply not the thing it is usually credited with.

What it provides is a constraint. Whatever a quantum state does, its centroid must move in a way consistent with the average force on it — so no quantum evolution can accelerate a centre of mass with no force present, and momentum conservation carries over intact. That is a strong statement about what quantum mechanics cannot do, and it is why quantum mechanics does not need a separate argument for the conservation of momentum.

It also gives the right answer whenever the force is linear, which covers more than the harmonic oscillator: a charge in a uniform electric field, a mass in a uniform gravitational field, and every free particle. In those cases the centroid’s motion is exactly classical for every state whatever, and calculations can be done by moving to a frame that follows it.

And it is the natural starting point for asking the harder question. The correction term identifies precisely what has to be small, in terms of quantities both theories possess; without the theorem there would be no clean way to say what “close to classical” means. The mistake is only in stopping at the first line.

The one exception, and why it is not one #

There is a state whose width does not change: the coherent state of a harmonic oscillator, a Gaussian of exactly the right width, which oscillates back and forth keeping its shape for ever.

The one exception is the harmonic oscillator, and it is not really an exception. Its ground state has a width it cannot go below, a coherent state is that same width displaced, and displacing it costs nothing in shape because the potential is exactly parabolic. So the packet moves without spreading and the centroid follows the classical path forever — which happens because the force is linear, which is the same reason as before rather than a new one.

It is tempting to read that as the classical limit realised. It is not, for the reason that has run through this whole essay: the harmonic oscillator’s force is linear, so the theorem’s correction vanishes for any state and the coherent state’s constancy of width is a separate bonus rather than the mechanism.

The coherent state is nevertheless the right object for a different purpose. It is the state a laser produces, it minimises the uncertainty product, and its expectation values follow the classical equations exactly — which is why the electromagnetic field of a laser can be treated as a classical wave with confidence, and why that confidence does not extend to a field in any nonlinear medium.

What was actually computed here #

The two figures are numerical solutions rather than illustrations, and the method is worth stating because it decides what the agreement in the first one is evidence of.

The wavefunction is propagated by split-step Fourier: half a step of the potential applied as a phase in position, a full step of the kinetic term applied as a phase in wavenumber with a fast Fourier transform on either side, then the other half of the potential. The classical trajectory is integrated separately, by fourth-order Runge–Kutta, from the same starting position and momentum, and nothing is shared between the two computations except the potential.

Three things are checked. The norm of the wavefunction is conserved to better than a part in $10^9$, which is what says the propagation is unitary and that anything read off it means something. In the harmonic case the two curves agree to about $10^{-5}$ of the amplitude, which is the split-step integrator’s own error over two periods and not a physical gap — halving the step quarters it. And in the quartic case the theorem is checked directly by comparing $\mathrm{d}\langle p\rangle/\mathrm{d}t$ against $\langle -V'\rangle$, which agree throughout while the classical comparison fails.

That last check is the important one. Without it the quartic figure would show two curves diverging and would not distinguish “the theorem fails here” from “the theorem is being misread here”, and those are entirely different claims.

Where it stops #

None of this is about measurement. Every statement above concerns the evolution of a state under the Schrödinger equation, with no observation anywhere. What a measurement does to a state is a separate matter and is not covered by Ehrenfest’s theorem, which is a statement about unitary evolution and says nothing about what happens when something is looked at.

Expectation values are not the whole state. Two very different states can share the same $\langle x \rangle$ and $\langle p \rangle$ — a narrow packet and a pair of packets on either side, for instance — and the theorem treats them identically. Knowing that the averages follow classical equations is therefore compatible with the state being nothing like a classical particle, which is the barrier-top case in general form.

The comparison is with one classical trajectory and there could be several. Setting the initial classical position and momentum to the quantum expectation values is a choice, and a defensible one; setting them to the most probable values, or to the values of the classical orbit with the same energy, gives different trajectories to compare against. For a narrow packet all these agree and the choice does not matter. For a wide one they do not, and the size of the disagreement between the possible comparisons is itself a measure of how meaningless the comparison has become.

And a “classical trajectory” is being assumed to exist. For the comparison to mean anything there must be a classical solution to compare with, which requires the classical problem to be well posed. Where it is not — a classical system whose trajectory is not unique, or one whose equations are singular — the question of correspondence has to be asked differently.

Where it stops is worth distinguishing from the usual account. The classical limit of a bound system is normally described as the limit of large quantum number, and the criterion in this essay is a different one: it is about the width of a state against the curvature of the potential it sits in. The two agree in the cases where both apply, and only the second says anything about how long the agreement lasts.

The theorem’s sibling, which is stronger #

There is a related result that does what Ehrenfest’s theorem is usually asked to do, and comparing them shows what was missing.

The Wigner function is a distribution over phase space built from the wavefunction, and its evolution equation is the classical Liouville equation plus a series of correction terms involving higher derivatives of the potential — the first of which is third order, with $\hbar^2$ in front of it. For a potential that is at most quadratic every correction vanishes identically, and the quantum distribution evolves exactly as a classical one would.

That is a much stronger statement than the theorem about centroids, because it is about the whole distribution rather than about two of its moments, and it makes the criterion explicit: the classical limit is good when the corrections involving the third and higher derivatives of the potential are small compared with the leading term, over the region the state occupies. It is the same condition this essay derived from the force, arrived at without averaging anything away, and it is where a careful treatment of the classical limit starts.

The ladder from here #

Later rungs on this anchor: the WKB approximation, which is the classical limit made into a calculational method rather than a check; quantum revivals, where a spread packet in a bound anharmonic potential reassembles after a long time and the correspondence returns periodically; the Wigner function, which makes the comparison between quantum and classical phase-space distributions directly and shows exactly which term in the evolution has no classical counterpart; and decoherence with its timescales, which is what actually supplies the classical world the theorem is often credited with.

The neighbouring ladders are where the quantum picture hands back the old one, which is the correspondence principle stated carefully; the packet that will not keep its shape, which is the spreading this argument depends on; and the motion that cannot be stopped, where the width that limits everything here has a floor.