Ehrenfest WKB Classical Limits

Ehrenfest equations and WKB asymptotics explain particular forms of quantum–classical correspondence. They do not assert that every quantum state becomes one classical trajectory. The semiclassical correspondence guide develops the controlled approximations; here the question is what they imply about motion and outcomes.

Required background. Density operators: Use density operators to calculate probabilities and distinguish coherent from incoherent alternatives.

Ehrenfest is an equation for means

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

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

The canonical theorem states its operator assumptions. Replacing the last mean by $-V'(\langle x\rangle)$ requires a sufficiently narrow distribution or a force linear in position.

A symmetric superposition of packets far to the left and right can have $\langle x\rangle=0$ even when almost no probability lies near zero. The mean is then a poor description of a typical recorded position. Exact mean equations do not repair this mismatch.

WKB describes phase and amplitude asymptotics

In a classically allowed one-dimensional region, a WKB branch has the form

$$\psi(x)\approx\frac{C}{\sqrt{p(x)}} \exp\!\left[\frac{i}{\hbar}\int^x p(x')\,dx'\right],$$

where $p(x)=\sqrt{2m(E-V(x))}$. The phase follows the classical action, while the amplitude reflects flux conservation. The approximation requires slow variation on the local wavelength scale and fails near ordinary turning points without connection formulas.

A stationary standing wave combines branches moving in opposite directions. It need not describe one localized moving particle. Classical trajectories organize the asymptotics of the wavefunction without uniquely selecting a trajectory ontology.

A coarse-resolution example

In a one-dimensional box of length $L$, a high-energy eigenstate has density

$$|\psi_n(x)|^2=\frac1L\left[1-\cos\left(\frac{2n\pi x}{L}\right)\right].$$

The oscillations do not vanish pointwise as $n$ grows. At fixed detector resolution, the measured probabilities approach those of the classical uniform spatial distribution. The canonical box correspondence example derives this result and states the observable class for which the error tends to zero.

The limiting statement is therefore weak or coarse-grained convergence for an observable class. It is not a literal disappearance of the wavefunction’s nodes at finite $n$.

Correspondence and definite outcomes

Semiclassical methods explain spectra, phases, packet motion, and smoothed probabilities under controlled conditions. Decoherence can additionally suppress interference between alternative records. Neither calculation by itself selects one unique global outcome from a unitary superposition.

The classical transition combines these mechanisms while keeping their claims distinct. A complete statement names the small parameter, fixed physical scales, state family, measured observables, error, and time interval.

Exercises

A fixed-resolution detector cannot distinguish a sufficiently high-energy box eigenstate from a uniform spatial distribution within its stated tolerance. Assess three conclusions: the density becomes uniform at every point; a detector whose resolution improves with $n$ must also see uniformity; the particle has thereby acquired one selected classical trajectory. Use the density at $x=L/2$ as a check on the first claim.

Solution

None of the conclusions follows. At $x=L/2$, the density is $[1-(-1)^n]/L$: it alternates between zero and $2/L$ rather than converging to $1/L$. Agreement at fixed resolution concerns averaged probabilities, not pointwise equality.

A detector whose resolution shrinks on the oscillation scale $L/n$ changes the observable family as the limit is taken. The fixed-resolution convergence result supplies no guarantee for that experiment; suitably placed narrow bins can continue to resolve nodes and peaks. Finally, even exact agreement with a classical position distribution would specify a statistical prediction, not one actual position history. A trajectory claim requires a dynamics, a state or conditional-record model, and a stated interpretation of its individual paths.

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315–397 (1972), doi:10.1088/0034-4885/35/1/306.
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007).