Classical Limit and Correspondence

Quantum predictions can agree with classical mechanics in suitable regimes, but different questions require different approximations. A wave packet’s center may follow a classical orbit, an environment may suppress local interference, or a coarse measurement may miss fine quantum structure. This chapter explains what each statement does—and does not—establish.

When does a packet center obey Newton’s equation?

For $H=P^2/(2m)+V(X)$ on the line, with suitable states and domains, Ehrenfest’s theorem gives

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

The second equation contains the average force, not generally the force at the average position. Newtonian evolution for the center follows exactly for a quadratic potential because its force is linear. For a smooth anharmonic potential, a narrow-packet expansion gives

$$\langle V'(X)\rangle =V'(\bar x)+\frac12 V^{(3)}(\bar x)(\Delta X)^2+\cdots, \qquad \bar x=\langle X\rangle.$$

Packet width and higher moments therefore matter. Even exact classical motion of a mean does not imply that the full quantum state is a point particle.

Semiclassical methods instead organize phases using a small dimensionless ratio such as $\hbar/S$, where $S$ is a relevant action scale. Decoherence describes suppression of selected off-diagonal terms in a reduced state. These mechanisms can work together, but none is simply the substitution $\hbar=0$ into every formula.

Reading sequence

Question	Article	Main distinction
What must quantum theory recover?	Correspondence Principle	empirical regime requirement versus hidden trajectory claim
When do expectation values look classical?	Ehrenfest Theorem Overview	exact mean equations versus approximate closure
Which mechanisms contribute to classical behavior?	Classical Limit	family of regimes versus one substitution
What does semiclassical approximation retain?	Semiclassical Limit Overview	asymptotic quantum phase versus exact classical theory
How does an environment suppress coherence?	Decoherence Preview	reduced classical appearance versus outcome selection
Why are quantization and classical recovery different?	Quantization vs Classical Limit	construction direction versus approximation direction
Which slogans should be rejected?	Common Misstatements About the Classical Limit	qualified mechanisms versus overclaiming

Follow the mechanism

Start with correspondence and the Ehrenfest example. Then choose the semiclassical route for WKB and stationary phase, or the decoherence route for a system coupled to an environment. In either case specify the states, observables, times, and accuracy you want to describe.

Try it. For $V(x)=\lambda x^4$, write the difference between the mean force and the force at $\bar x$. Let $\mu_3=\langle(X-\bar x)^3\rangle$.

Solution

Since $\langle X^3\rangle=\bar x^3+3\bar x(\Delta X)^2+\mu_3$, the difference is

$$-4\lambda\langle X^3\rangle+4\lambda\bar x^3 =-12\lambda\bar x(\Delta X)^2-4\lambda\mu_3.$$

A symmetric packet has $\mu_3=0$, but its width still affects the mean force when $\bar x\ne0$. A centered symmetric packet can have zero mean force while remaining spread out.

References

• M. V. Berry, “Regular and irregular semiclassical wavefunctions,” Journal of Physics A 10, 2083–2091, 1977.
• N. Bohr, The Theory of Spectra and Atomic Constitution, Cambridge University Press, 1922.
• P. Ehrenfest, “Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik,” Zeitschrift für Physik 45, 455–457, 1927.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• E. Joos et al., Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed., Springer, 2003.
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715–775, 2003, doi:10.1103/RevModPhys.75.715.