Dispersion and Classical Limit

A free wave packet looks classical when its center follows the classical trajectory and its width remains small compared with the length scale being resolved. Dispersion is the mechanism that can spoil this picture: different momentum components move with different group velocities, so the packet spreads.

The free-particle case is unusually clean because the center motion is exactly classical:

$$\langle x\rangle(t) = \langle x\rangle(0) +\frac{\langle p\rangle}{m}t.$$

The classical-limit question is therefore not whether the center obeys Newton’s law. It does. The question is whether the state remains localized enough for that center to represent the observed motion.

Free Packet In Momentum Space

With the site momentum convention, a normalizable free packet can be written as

$$\psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) \exp\left[ \frac{i}{\hbar}px -\frac{i}{\hbar}\frac{p^2}{2m}t \right]\,dp.$$

If $\phi(p)$ is concentrated near $p_0$, write $p=p_0+q$. The energy expands as

$$E(p_0+q) = \frac{p_0^2}{2m} +\frac{p_0}{m}q +\frac{q^2}{2m}.$$

The first term gives an overall phase. The linear term translates the packet with velocity

$$v_0=\frac{p_0}{m}.$$

The quadratic term is dispersion. It changes the relative phases across the momentum distribution and therefore changes the packet shape.

Narrow Momentum Distribution

A packet has a well-defined classical velocity when its momentum spread is small compared with its mean momentum:

$$\Delta p\ll \lvert p_0\rvert.$$

Then the velocity spread

$$\Delta v=\frac{\Delta p}{m}$$

is small compared with

$$\lvert v_0\rvert=\frac{\lvert p_0\rvert}{m}.$$

This condition is about directional and velocity sharpness. It is not enough by itself. A packet can have a narrow momentum distribution and still be very broad in position. Classical-looking particle motion also requires the spatial width to be small compared with the relevant observation scale.

Short-Time Propagation

Over a time $t$, different velocity components separate by a distance of order

$$\Delta x_{\mathrm{disp}} \sim \Delta v\,t = \frac{\Delta p}{m}t.$$

If this is small compared with the initial width or with the measurement resolution, the packet looks approximately rigid. A practical short-time condition is

$$t\ll t_{\mathrm{disp}},$$

where $t_{\mathrm{disp}}$ is the characteristic time on which the packet width changes appreciably.

For an initially unchirped minimum-uncertainty Gaussian,

$$t_{\mathrm{disp}} \sim \frac{2m\sigma_x^2}{\hbar}.$$

This formula encodes the main scaling:

• larger mass slows dispersion;
• wider initial packets disperse more slowly;
• sharper localization disperses more quickly.

The exact Gaussian formula is derived in Gaussian Wave Packets and interpreted in Wave Packet Spreading.

Large Mass Limit

For fixed initial width $\sigma_x$, the minimum momentum spread is

$$\Delta p_{\min} = \frac{\hbar}{2\sigma_x}.$$

This lower bound does not contain the mass. The velocity spread does:

$$\Delta v_{\min} = \frac{\Delta p_{\min}}{m} = \frac{\hbar}{2m\sigma_x}.$$

Thus large mass suppresses the rate at which different momentum components separate in space. This is one reason macroscopic packets can remain effectively localized over ordinary times and length scales even though the uncertainty relation still holds exactly.

The large-mass limit is not magic. If one waits long enough, or asks for enough spatial resolution, dispersion can still matter. Classical behavior is always relative to scales.

Ehrenfest Center Motion

For a free particle, Ehrenfest theorem gives

$$\frac{d}{dt}\langle x\rangle = \frac{\langle p\rangle}{m}, \qquad \frac{d}{dt}\langle p\rangle=0.$$

Therefore

$$\langle x\rangle(t) = x_0+\frac{p_0}{m}t$$

when $x_0=\langle x\rangle(0)$ and $p_0=\langle p\rangle$.

This exact center motion is necessary for a classical-particle approximation, but not sufficient. A broad or split wavefunction may have a mean position that is not where a measurement is likely to find the particle. The Ehrenfest Theorem Overview explains this limitation for general potentials.

Spatial Resolution And Coarse Graining

Let $L_{\mathrm{obs}}$ be the spatial resolution or length scale relevant to the question. A free packet is particle-like over the time interval of interest if

$$\Delta x(t)\ll L_{\mathrm{obs}}.$$

For scattering, $L_{\mathrm{obs}}$ might be a detector resolution, a barrier width, or the distance over which an asymptotic beam is considered well collimated. For a classical trajectory comparison, it might be the scale on which the trajectory is plotted or measured.

This scale dependence matters. The same packet can be classical-looking for a coarse detector and visibly quantum for a fine interferometric measurement.

De Broglie Wavelength And Envelope Scale

A packet with central momentum $p_0$ has de Broglie wavelength

$$\lambda_{\mathrm{dB}} = \frac{2\pi\hbar}{\lvert p_0\rvert}.$$

For a semiclassical packet, the carrier oscillations are usually much shorter than the envelope scale:

$$\lambda_{\mathrm{dB}}\ll \Delta x.$$

Equivalently, the packet contains many phase oscillations under its envelope. This is compatible with a narrow relative momentum spread:

$$\Delta p\ll \lvert p_0\rvert.$$

When the de Broglie wavelength is comparable to the entire packet or apparatus scale, wave effects are not small corrections; they are the main description.

Stationary Phase View

The free packet integral has rapidly oscillating phase

$$\Phi(p) = px-\frac{p^2}{2m}t.$$

The stationary phase condition is

$$\frac{d\Phi}{dp} = x-\frac{p}{m}t =0.$$

Thus the dominant momentum contributing to position $x$ at time $t$ satisfies

$$x=\frac{p}{m}t,$$

which is the classical free trajectory from the origin. Stationary phase is one route from wave mechanics to classical propagation: when phases oscillate rapidly, contributions away from classical stationary paths cancel strongly.

This idea becomes more powerful in WKB and semiclassical propagator methods. The broader map is in Semiclassical Limit Overview, WKB Approximation, and Semiclassical Propagator.

A Practical Classicality Checklist

For a free wave packet over a chosen time interval, check:

• Is $\Delta p\ll\lvert p_0\rvert$, so the velocity is well defined?
• Is $\Delta x(t)\ll L_{\mathrm{obs}}$, so the packet is localized at the resolution of interest?
• Is $t\ll t_{\mathrm{disp}}$, or otherwise is the spreading negligible for the question?
• Is $\lambda_{\mathrm{dB}}\ll\Delta x$, so many carrier wavelengths fit under the envelope?
• Is the state single-peaked rather than a superposition of widely separated packets?
• Are interference fringes irrelevant at the chosen coarse graining?

If these answers are yes, the packet center can often be treated as a classical particle trajectory. If not, the wave description is not optional.

Common Mistakes

• Saying the classical limit is simply $\hbar\to0$ without specifying scales.
• Confusing exact Ehrenfest center motion with full classical particle behavior.
• Assuming a narrower packet is always more classical; narrow packets often spread faster.
• Ignoring the momentum-spread condition $\Delta p\ll\lvert p_0\rvert$.
• Treating de Broglie wavelength as irrelevant once the packet center moves classically.
• Forgetting that coarse graining and detector resolution are part of practical classicality.

Where This Is Used

• Wave Packet Spreading gives the width evolution and spreading time.
• Group Velocity and Phase Velocity explains the envelope velocity.
• Minimum-Uncertainty Wave Packets explains the Gaussian lower bound on $\Delta x\,\Delta p$.
• Free-Particle Propagator: First Encounter gives the kernel view of free propagation.
• Ehrenfest Theorem Overview explains the center-of-packet bridge to Newtonian motion.
• Wave Packets and Classical Trajectories extends the free-packet checklist to external potentials, coherent states, and split branches.
• Correspondence Principle and Semiclassical Limit Overview give broader classical-limit context.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.

Exercises

1. A Gaussian packet has initial width $\sigma_x$ and mass $m$. How does $t_{\mathrm{disp}}$ change if the mass is multiplied by $10$?

Solution

For an initially unchirped minimum-uncertainty Gaussian,

$$t_{\mathrm{disp}} \sim \frac{2m\sigma_x^2}{\hbar}.$$

Multiplying $m$ by $10$ multiplies $t_{\mathrm{disp}}$ by $10$.

2. A packet has mean momentum $p_0$ and spread $\Delta p=0.01\lvert p_0\rvert$. What does this say about its velocity spread?

Solution

The mean speed scale is

$$\lvert v_0\rvert=\frac{\lvert p_0\rvert}{m},$$

and the velocity spread is

$$\Delta v=\frac{\Delta p}{m} = 0.01\frac{\lvert p_0\rvert}{m} = 0.01\lvert v_0\rvert.$$

The packet has a one-percent relative velocity spread.

3. Use stationary phase to recover the classical relation between $x$, $p$, and $t$ for a free particle.

Solution

The free-particle phase in the momentum integral is

$$\Phi(p)=px-\frac{p^2}{2m}t.$$

Stationary phase requires

$$\frac{d\Phi}{dp} = x-\frac{p}{m}t =0.$$

Thus

$$x=\frac{p}{m}t,$$

which is the classical free-particle trajectory from the origin with velocity $p/m$.

4. Why is a narrow spatial packet not automatically more classical?

Solution

A narrow spatial packet has a large momentum spread by the uncertainty relation. The velocity spread is $\Delta v=\Delta p/m$, so its components separate more quickly. Unless the mass is large or the time interval is short, sharper localization can make dispersion more important, not less.