Free Motion and Wave Packets

A free particle has no force acting on it, yet a localized quantum packet can change shape. This chapter explains the contrast between the uniform motion of its center and the spreading or focusing caused by its momentum distribution. Plane waves supply the basis; normalizable superpositions describe localized preparations.

A moving center and a changing width

On the full line, $H=P^2/(2m)$ has momentum eigenfunctions proportional to $e^{ipx/\hbar}$ and energy $E=p^2/(2m)$. Opposite momenta have equal energy but opposite currents. A plane wave is a generalized eigenstate, not a unit-norm state on the line.

An initially unchirped Gaussian with position standard deviation $\sigma_0$, mean position $x_0$, and mean momentum $p_0$ evolves with

$$\langle X\rangle(t)=x_0+\frac{p_0t}{m},\qquad \sigma_x(t)=\sigma_0\sqrt{1+\left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}.$$

Its center follows classical free motion exactly while its width grows. This particular width formula assumes zero initial position–momentum covariance. A negatively correlated packet can initially contract before expanding; free evolution does not require every packet to widen monotonically from every chosen starting time.

The general connection is Fourier superposition: different momentum components acquire different phases under evolution. The Gaussian calculation makes those phases explicit, while spreading develops the covariance formula.

Reading sequence

Page	Central question
Free Particle	What spectrum and current follow from $\hat H=\hat p^2/(2m)$?
Plane Waves and Delta Normalization	How are non-normalizable continuum eigenstates used consistently?
Momentum Eigenstates	How do plane waves represent definite momentum and translations?
Gaussian Wave Packets	What is the exact localized Gaussian state and its free evolution?
Wave Packet Spreading	How do momentum spread and covariance determine spatial width?
Group Velocity and Phase Velocity	Which velocity tracks the packet envelope?
Free-Particle Propagator: First Encounter	How does a kernel evolve arbitrary initial data?
Minimum-Uncertainty Wave Packets	What does saturation of the uncertainty relation actually imply?
Dispersion and Classical Limit	Which scales make a spreading packet approximate a classical particle?

Choose the next step

Read the free particle and continuum normalization before Gaussian packets. Use group velocity to interpret the envelope’s motion, then study spreading and minimum uncertainty. The propagator gives another way to evolve arbitrary initial data.

Exercises

For the Gaussian above, how does the time needed for its width to reach $\sqrt2\sigma_0$ change if the initial width doubles at fixed mass?

Solution

The time is $t_s=2m\sigma_0^2/\hbar$, so doubling $\sigma_0$ multiplies $t_s$ by four. Sharper initial localization requires a broader momentum distribution and spreads more rapidly in this unchirped family.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.