Emergence of Classical Trajectories

A classical trajectory is an approximation to the evolution of suitably localized observables or to a conditioned measurement record. It does not follow from suppression of interference alone. The approximation must control wave-packet spreading, force variation, measurement backaction, and the time interval over which errors remain small.

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

Localized means and smooth forces

The Ehrenfest theorem supplies equations for expectation values. Approximating the mean force by the force at the mean requires a narrow distribution relative to the length scale on which the force changes.

For a quartic potential $V(x)=gx^4/4$, the exact mean force is $-g\langle x^3\rangle$. Writing $x=\bar x+\delta x$ gives

$$\langle x^3\rangle=\bar x^3+3\bar x\,\sigma_x^2+ \langle(\delta x)^3\rangle.$$

Even a symmetric packet has a variance correction. A broad packet need not follow the Newtonian orbit generated by $-g\bar x^3$.

Spreading sets a time window

For an initially unchirped minimum-uncertainty free Gaussian,

$$\sigma_x^2(t)=\sigma_x^2(0)+ \frac{\hbar^2t^2}{4m^2\sigma_x^2(0)}.$$

A large mass slows spreading, but making the initial packet arbitrarily narrow increases its momentum spread. The condition $\sigma_x(t)\ll L$ must hold for the length scale $L$ relevant to the desired trajectory resolution.

The packet center can obey an exact classical equation while the packet itself spreads. Following the mean is therefore weaker than representing the state by a point.

Monitoring localizes and perturbs

Continuous monitoring can maintain localization in a state conditioned on a measurement record. It also introduces fluctuations and momentum diffusion. Schematically, a classical stochastic approximation may take the form

$$dx=\frac p m\,dt,\qquad dp=-V'(x)\,dt-\gamma p\,dt+\sqrt{2D_p}\,dW_t.$$

This Langevin equation is an effective model, not a universal consequence of observing position. Its damping and diffusion coefficients must come from a specified environment or measurement model.

A smooth approximately deterministic trajectory requires the accumulated noise to be small relative to the resolution scale. If the noise is appreciable, a classical stochastic process may still be valid even though a single Newtonian orbit is not.

Conditional and unconditioned descriptions

A conditioned state follows one observed record; averaging over records yields an unconditioned density operator that may be broad. Different measurement schemes can produce different unravelings of the same reduced master equation.

Thus a stochastic trajectory representation does not automatically establish an observer-independent hidden path. That further claim depends on the interpretation or physical model. The distinction is essential when comparing monitoring theory with Bohmian trajectories or objective collapse paths.

Exercises

For the free Gaussian, find the time at which its position variance doubles. How does that time scale with mass and initial width?

Solution

Set $\hbar^2t^2/(4m^2\sigma_0^2)=\sigma_0^2$, giving $t=2m\sigma_0^2/\hbar$. The time increases linearly with mass and quadratically with initial width. Sharper initial localization shortens the spreading time.

References

• T. Bhattacharya, S. Habib, and K. Jacobs, “Continuous quantum measurement and the emergence of classical chaos,” Physical Review Letters 85, 4852–4855 (2000), doi:10.1103/PhysRevLett.85.4852.
• W. H. Zurek, S. Habib, and J. P. Paz, “Coherent states via decoherence,” Physical Review Letters 70, 1187–1190 (1993), doi:10.1103/PhysRevLett.70.1187.