Wave Packets and Classical Trajectories

A wave packet behaves particle-like when its quantum state remains concentrated around a classical phase-space path at the resolution and for the duration relevant to the problem. The center must move approximately classically, but that is only one requirement: the packet must also avoid excessive spreading, distortion, splitting, tunneling, and observable interference.

The exact free Gaussian, group-velocity derivation, and spreading formulas have canonical homes in Gaussian Wave Packets, Group Velocity and Phase Velocity, and Wave Packet Spreading. This page synthesizes them into a trajectory criterion and extends the discussion to external potentials.

For a packet assembled from one isolated Bloch band, Semiclassical Dynamics of Bloch Electrons owns crystal-momentum evolution and weak electric- and magnetic-field motion. The generic localization, spreading, splitting, and one-center trajectory criteria remain here.

Localized States Are Not Classical Points

A classical one-particle state is a point

$$z_{\rm cl}=(x_{\rm cl},p_{\rm cl})$$

in phase space. A quantum packet instead has finite spreads

$$\sigma_x^2 = \langle(x-\langle x\rangle)^2\rangle,$$

and

$$\sigma_p^2 = \langle(p-\langle p\rangle)^2\rangle.$$

With symmetrized covariance

$$C_{xp} = \frac{1}{2} \left\langle \delta x\,\delta p + \delta p\,\delta x \right\rangle,$$

the Robertson–Schrödinger inequality is

$$\sigma_x^2\sigma_p^2 - C_{xp}^2 \geq \frac{\hbar^2}{4}.$$

The packet therefore occupies an irreducible phase-space ellipse or more complicated distribution. A classical trajectory approximation replaces that finite quantum structure by a narrow tube around one path, not by an exact point with simultaneous sharp position and momentum.

If $L_x$ and $P_p$ are the resolved position and momentum scales, useful localization conditions are

$$\frac{\sigma_x(t)}{L_x} \ll1, \qquad \frac{\sigma_p(t)}{P_p} \ll1.$$

The scales and time interval must be stated. The same packet can be particle-like for a coarse detector and strongly wave-like in an interferometer.

Fourier Construction and Group Motion

A one-dimensional packet can be written

$$\psi(x,t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dk\, a(k)e^{i[kx-\omega(k)t]}.$$

Suppose $a(k)$ is concentrated near $k_0$. Set

$$\kappa=k-k_0$$

and expand the dispersion relation:

$$\begin{aligned} \omega(k) &= \omega_0 + \omega_0'\kappa + \frac{1}{2}\omega_0''\kappa^2 \\ &\quad+ \frac{1}{6}\omega_0'''\kappa^3 + \cdots. \end{aligned}$$

The constant term supplies an overall carrier phase. The linear term translates the envelope with group velocity

$$v_g = \omega_0' = \left. \frac{d\omega}{dk} \right|_{k_0}.$$

If the higher terms are negligible, the envelope is approximately

$$\psi(x,t) \approx e^{i(k_0x-\omega_0t)} A(x-v_gt),$$

where $A$ is the initial slowly varying envelope up to a convention-dependent phase.

This is the ray or group-trajectory picture: the packet peak follows

$$x_c(t) \approx x_c(0)+v_gt.$$

It requires a narrow spectrum and a time interval short enough that dispersion has not reshaped the envelope appreciably.

Dispersion and the Rigid-Packet Approximation

Across a spectral width $\Delta k$, the quadratic phase accumulated over time $t$ is of order

$$\epsilon_{\rm disp}(t) \sim \left\lvert\omega_0''\right\rvert (\Delta k)^2t.$$

An approximately rigid-envelope regime requires

$$\epsilon_{\rm disp}(t) \ll1,$$

up to order-one factors set by the packet convention. The same curvature gives a spread of group velocities

$$\Delta v_g \sim \left\lvert\omega_0''\right\rvert \Delta k.$$

For a nonrelativistic free particle,

$$\omega(k) = \frac{\hbar k^2}{2m},$$

so

$$v_g = \frac{\hbar k_0}{m} = \frac{p_0}{m}, \qquad \omega'' = \frac{\hbar}{m}.$$

The center moves at the classical velocity, while nonzero $\omega''$ causes spreading. A linear dispersion relation would transport the envelope without this quadratic dispersion, although higher-order terms, interactions, boundaries, or multiple branches could still distort it.

Free Gaussian Benchmark

For an initially unchirped minimum-uncertainty Gaussian,

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

and

$$\sigma_x(t) = \sigma_0 \sqrt{ 1+ \left( \frac{\hbar t}{2m\sigma_0^2} \right)^2 }.$$

Define

$$t_{\rm sp} = \frac{2m\sigma_0^2}{\hbar}.$$

For $t\ll t_{\rm sp}$, the width changes little. For $t$ comparable to or larger than $t_{\rm sp}$, the packet no longer resembles a rigid localized object at its original resolution.

The general free-packet variance is

$$\begin{aligned} \sigma_x^2(t) &= \sigma_x^2(0) + \frac{2t}{m}C_{xp}(0) \\ &\quad+ \frac{t^2}{m^2}\sigma_p^2(0). \end{aligned}$$

An initial negative covariance can focus a packet temporarily, but the momentum-spread term dominates at sufficiently long times. Unitary normalization does not prevent spatial spreading.

External Potentials

Let

$$q(t)=\langle x\rangle, \qquad \xi=x-q(t).$$

Expand a smooth potential over the packet:

$$\begin{aligned} V(q+\xi) &= V(q) + V'(q)\xi + \frac{1}{2}V''(q)\xi^2 \\ &\quad+ \frac{1}{6}V'''(q)\xi^3 + \cdots. \end{aligned}$$

The terms play different roles:

• $V(q)$ contributes a phase along the packet center;
• $V'(q)$ accelerates the center;
• $V''(q)$ focuses, defocuses, squeezes, or rotates the covariance;
• $V'''(q)$ and higher derivatives generate non-Gaussian distortion and couple the center to higher moments.

Ehrenfest’s theorem gives

$$m\ddot q = -V'(q) - \frac{1}{2}V'''(q)\sigma_x^2 - \cdots.$$

Thus a packet follows an approximate Newtonian path when it remains narrow relative to the force-variation scale and the omitted moments remain small. Ehrenfest Theorem Revisited develops this closure condition quantitatively.

A useful local estimate for neglecting cubic distortion over an interval $t$ is

$$\frac{ \left\lvert V'''(q)\right\rvert \sigma_x^3t }{\hbar} \ll1,$$

provided the potential derivatives and width do not vary too much during that interval. This is a diagnostic, not a universal theorem; caustics, turning points, and unstable flows require more careful semiclassical analysis.

Three Instructive Cases

Constant force

For

$$V(x)=-Fx,$$

the center obeys

$$q(t) = q_0+\frac{p_0}{m}t + \frac{F}{2m}t^2.$$

The potential is linear, so it does not add force curvature across the packet. Its width evolves as in free motion, while the whole packet accelerates. Classical center motion can therefore be exact even though quantum spreading continues.

Harmonic coherent state

For a harmonic oscillator, a coherent state remains a Gaussian with fixed covariance and a center that follows the exact classical ellipse. This is the cleanest single-trajectory packet model. Coherent States owns the Hilbert-space construction, Coherent States in Phase Space owns the Gaussian Wigner ellipse, and Coherent-State Dynamics owns exact harmonic and driven evolution.

A squeezed oscillator state also has exactly classical first moments, but its covariance oscillates. Fixed shape is a special property of coherent states, not a consequence of Ehrenfest’s theorem alone.

Scattering and splitting

An incoming packet incident on a barrier can separate into reflected and transmitted packets. At late times there may be two well-localized outgoing branches, each associated with a different approximate path. The total state is not represented by one trajectory, and its mean position can lie between the branches.

Wave Packets and Scattering owns the construction from stationary amplitudes and the late-time reflected and transmitted probabilities.

Phase-Space Picture

For Gaussian states, the first moments form a center vector

$$\mathbf d = \begin{pmatrix} \langle x\rangle \\ \langle p\rangle \end{pmatrix},$$

and the second moments form a covariance matrix

$$\Sigma = \begin{pmatrix} \sigma_x^2 & C_{xp} \\ C_{xp} & \sigma_p^2 \end{pmatrix}.$$

Quadratic Hamiltonians move $\mathbf d$ by the classical affine symplectic flow and transform $\Sigma$ by the corresponding linear map. The center is the candidate trajectory; the covariance is the tube around it.

For nonquadratic Hamiltonians, the Wigner function can shear, fold, and develop interference fringes or negative regions. One center and one covariance ellipse then cease to describe the full state. The canonical phase-space treatment is Gaussian States and Wigner Functions.

Coarse-Grained Trajectories

Experiments rarely resolve every quantum oscillation. Suppose position is measured with resolution $L_{\rm obs}$ and momentum with resolution $P_{\rm obs}$. If

$$\sigma_x(t)\ll L_{\rm obs}, \qquad \sigma_p(t)\ll P_{\rm obs},$$

then replacing the packet by its center can preserve the reported data even though the exact state remains extended. This is an operational trajectory approximation.

Coarse graining does not destroy quantum structure in principle. A later interference experiment can reveal phases that a position tracker ignored. The approximation is tied to a family of observables and a resolution, not to a declaration that the state has become a classical point.

Limits of Trajectory Language

Several distinct ideas are called a “quantum trajectory”:

• the path followed by a packet center in a semiclassical approximation;
• one branch or ray in a WKB or geometric-optics construction;
• a stochastic conditioned state in continuous-measurement theory;
• a Bohmian configuration trajectory in a specific interpretation;
• a path variable integrated over in a path integral.

They are not interchangeable. This page uses trajectory only for an approximate path representing a localized packet center or a resolved branch.

A single such path fails when

• the state is broad on the observation scale;
• the packet disperses or develops strong covariance;
• external forces vary appreciably across it;
• the state splits into reflected, transmitted, tunneled, or otherwise distinct branches;
• interference between branches affects later observables;
• tunneling reaches regions with no real classical connecting path;
• chaotic stretching magnifies the packet to macroscopic scales;
• environmental noise requires a stochastic rather than deterministic classical model.

Practical Checklist

Before replacing a packet by one trajectory, check

1. Is the state normalizable and initially single-peaked?
2. Are $\sigma_x/L_x$ and $\sigma_p/P_p$ small for the observables being resolved?
3. Does the group-velocity expansion remain accurate across the packet bandwidth?
4. Is $t$ short compared with the relevant spreading or distortion time?
5. Does the force vary little across the packet?
6. Are reflection, tunneling, caustics, and branch splitting negligible?
7. Is the mean located in a region of appreciable probability?
8. Is the required classical model deterministic, statistical, or stochastic?

If any answer changes during evolution, the trajectory approximation must be reassessed rather than carried forward automatically.

Common Mistakes

• Treating a plane wave as a localized particle trajectory.
• Identifying phase velocity with packet velocity.
• Assuming group velocity guarantees a rigid envelope.
• Calling a narrower initial packet more classical without checking its momentum spread.
• Tracking only $\langle x\rangle$ while ignoring width and covariance.
• Treating a split packet’s mean as one physical branch.
• Assuming a positive Gaussian Wigner function is an exact classical joint probability.
• Using “quantum trajectory” without declaring which meaning is intended.
• Forgetting that detector resolution is part of a practical classicality claim.

References

• E. J. Heller, “Time-dependent approach to semiclassical dynamics,” Journal of Chemical Physics 62, 1544–1555, 1975, doi:10.1063/1.430620.
• G. A. Hagedorn, “Semiclassical quantum mechanics. I. The $\hbar\to0$ limit for coherent states,” Communications in Mathematical Physics 71, 77–93, 1980, doi:10.1007/BF01230088.
• R. G. Littlejohn, “The semiclassical evolution of wave packets,” Physics Reports 138, 193–291, 1986, doi:10.1016/0370-1573(86)90103-1.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315–397, 1972.
• R. W. Robinett, “Quantum wave packet revivals,” Physics Reports 392, 1–119, 2004.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.

Exercises

1. Derive the group velocity from the first-order dispersion expansion.

Solution

Keeping only the constant and linear terms,

$$\omega(k) \approx \omega_0+\omega_0'(k-k_0).$$

Insert this into the packet integral and set $\kappa=k-k_0$:

$$\begin{aligned} \psi(x,t) &\approx e^{i(k_0x-\omega_0t)} \frac{1}{\sqrt{2\pi}} \int d\kappa \\ &\quad\times a(k_0+\kappa) e^{i\kappa(x-\omega_0't)}. \end{aligned}$$

The integral is the initial envelope evaluated at $x-\omega_0't$. Therefore the envelope moves with

$$v_g=\omega_0'.$$

2. For a free particle, estimate how the velocity spread depends on $\Delta k$.

Solution

The group velocity is

$$v_g(k)=\frac{\hbar k}{m}.$$

Linearizing over the packet bandwidth gives

$$\Delta v_g \approx \left\lvert \frac{dv_g}{dk} \right\rvert \Delta k = \frac{\hbar}{m}\Delta k.$$

After time $t$, this produces a spatial separation of order $\Delta v_g t$, which is the scaling behind free-packet spreading.

3. Show that a constant force changes the center motion but not the variance relative to free evolution.

Solution

For $H=p^2/(2m)-Fx$, the Heisenberg solutions are

$$p(t)=p(0)+Ft,$$

and

$$x(t) = x(0) + \frac{p(0)}{m}t + \frac{F}{2m}t^2.$$

The last term is a scalar shift, so it cancels from $x(t)-\langle x(t)\rangle$. Therefore

$$\begin{aligned} \sigma_x^2(t) &= \sigma_x^2(0) + \frac{2t}{m}C_{xp}(0) \\ &\quad+ \frac{t^2}{m^2}\sigma_p^2(0), \end{aligned}$$

the same variance formula as for free motion.

4. For an unchirped minimum-uncertainty free Gaussian, choose the initial width that minimizes the final width at a fixed time $t\gt0$.

Solution

The final variance is

$$\sigma_x^2(t) = \sigma_0^2 + \frac{\hbar^2t^2} {4m^2\sigma_0^2}.$$

Set

$$y=\sigma_0^2, \qquad A=\frac{\hbar t}{2m}.$$

Then

$$\sigma_x^2(t)=y+\frac{A^2}{y}.$$

Differentiating with respect to $y$ gives a minimum at

$$y=A.$$

Thus

$$\sigma_0^2 = \frac{\hbar t}{2m},$$

and the minimum final variance is

$$\sigma_{x,\min}^2(t) = \frac{\hbar t}{m}.$$

Making the initial packet narrower than this increases momentum spread enough to make the final packet wider.

5. Why can two outgoing scattering packets admit two approximate trajectories while their superposition admits no single one?

Solution

After reflection and transmission separate spatially, each branch can be narrow around its own center and momentum, so each can be approximated by a different path. The total state contains both alternatives and possibly a physically relevant relative phase. Its overall mean can lie between the packets, where little probability is present. One path cannot encode both branch probabilities, their distinct motions, and any later interference.

6. Explain why the covariance determinant prevents an exact quantum phase-space point.

Solution

The uncertainty relation is

$$\det\Sigma = \sigma_x^2\sigma_p^2 - C_{xp}^2 \geq \frac{\hbar^2}{4}.$$

An exact phase-space point would require every entry of $\Sigma$ to vanish, giving $\det\Sigma=0$. That contradicts the lower bound for $\hbar\ne0$. A quantum packet can be narrow relative to macroscopic scales, but it cannot be a simultaneous position–momentum delta function.