Free Motion and Wave Packets

Free motion is the canonical laboratory for continuous spectra, generalized eigenstates, Fourier synthesis, localization, and dispersion. With no potential and no force, the center of a free packet follows the classical trajectory exactly, yet the packet generally spreads. That contrast separates mean motion from the full quantum state.

The mathematical theory of Fourier transforms and distributions remains canonical in the Mathematical Toolkit. General propagator theory and path integrals remain canonical in Quantum Dynamics. This chapter owns how those tools describe the free particle in coordinate-space wave mechanics.

The free Hamiltonian

On the full line, the one-dimensional free Hamiltonian is

$$\hat H = \frac{\hat p^2}{2m} = - \frac{\hbar^2}{2m} \frac{d^2}{dx^2}.$$

The stationary Schrödinger equation has plane-wave solutions

$$\psi_k(x) = A e^{ikx}, \qquad E(k) = \frac{\hbar^2k^2}{2m}.$$

The spectrum is continuous. For every positive energy there are two propagation directions, $k$ and $-k$. Energy therefore does not determine momentum in one-dimensional free motion.

Free Particle owns the Hamiltonian, stationary equation, spectrum, current, and first classical comparison.

Generalized states and normalization

A plane wave has constant magnitude on the full line, so it cannot satisfy ordinary square normalization. With the chapter’s momentum convention,

$$\langle x\vert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} \exp\left( \frac{ipx}{\hbar} \right),$$

and generalized momentum eigenstates satisfy

$$\langle p\vert p'\rangle = \delta(p-p').$$

This is not a statement that a plane wave is an ordinary vector of unit norm. It is distributional normalization within a continuous spectral resolution. Box normalization can regulate intermediate formulas by replacing the continuum with discrete momenta, but the box size must be removed consistently.

Plane Waves and Delta Normalization owns continuum completeness and the box-to-delta correspondence. Momentum Eigenstates owns the eigenvalue equation, momentum probability density, translation interpretation, and distinction between momentum and energy.

Localized states require superposition

A normalizable free state is assembled from momentum amplitudes:

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

The momentum density $\lvert\phi(p)\rvert^2$ is time independent for a free particle. Time evolution changes the relative phases of its momentum components, which changes the position-space shape.

A narrow packet centered at $p_0$ has envelope velocity

$$v_g = \frac{d\omega}{dk}\bigg\rvert_{k_0} = \frac{\hbar k_0}{m} = \frac{p_0}{m}.$$

For the same component, the phase velocity is

$$v_{\mathrm{ph}} = \frac{\omega}{k} = \frac{\hbar k}{2m} = \frac12v_g.$$

The phase fronts of one delocalized component are not the trajectory of a localized particle. Group Velocity and Phase Velocity makes this distinction precise.

Gaussian benchmark

A convenient minimum-uncertainty packet at $t=0$ is

$$\psi(x,0) = \left( \frac{1}{2\pi\sigma_x^2} \right)^{1/4} \exp\left[ - \frac{(x-x_0)^2}{4\sigma_x^2} + \frac{i}{\hbar}p_0(x-x_0) \right].$$

Here $\sigma_x$ is the initial position uncertainty, and

$$(\Delta p)_0 = \frac{\hbar}{2\sigma_x}, \qquad (\Delta x)_0(\Delta p)_0 = \frac{\hbar}{2}.$$

Under free evolution the probability density remains Gaussian. Its center and width are

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

and

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

Gaussian Wave Packets owns the exact state and momentum transform. Wave Packet Spreading owns the general covariance-based width formula and the physical mechanism.

Why spreading occurs

The free dispersion relation is quadratic:

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

Different momentum components therefore have different group velocities. A localized state necessarily contains a momentum range, so those components separate. The characteristic Gaussian spreading time is

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

Larger mass and broader initial width delay spreading. Sharper localization increases momentum spread and accelerates it. The uncertainty relation is preserved, but free evolution can build position–momentum correlations, so a packet that initially saturates the lower bound need not remain an uncorrelated minimum-width packet.

Minimum-Uncertainty Wave Packets distinguishes saturation from localization and follows those correlations.

Propagation as a kernel

The free propagator is

$$K(x,t;x',0) = \sqrt{ \frac{m}{2\pi i\hbar t} } \exp\left[ \frac{im(x-x')^2}{2\hbar t} \right], \qquad t\gt0.$$

It evolves initial data by

$$\psi(x,t) = \int_{-\infty}^{\infty} K(x,t;x',0)\psi(x',0)\,dx'.$$

The exponent contains the classical free-particle action, while the prefactor enforces normalization, composition, and the distributional initial condition. Free-Particle Propagator: First Encounter introduces this kernel without duplicating the general treatment in Propagator as a Kernel.

When motion looks classical

For every free state with finite first moments,

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

Classical-looking motion requires more than this exact center trajectory. The packet must remain narrow compared with the spatial resolution and dynamical length scales of interest. Useful conditions include

$$\Delta p \ll \lvert p_0\rvert, \qquad t \ll t_{\mathrm{spread}},$$

together with a position width small enough for the physical question. Classicality is therefore scale- and time-dependent, not a declaration that the state has become a point in phase space.

Dispersion and Classical Limit owns this controlled comparison and links it to Ehrenfest and semiclassical reasoning.

Reading route

1. Free Particle
2. Plane Waves and Delta Normalization
3. Momentum Eigenstates
4. Gaussian Wave Packets
5. Group Velocity and Phase Velocity
6. Wave Packet Spreading
7. Minimum-Uncertainty Wave Packets
8. Free-Particle Propagator: First Encounter
9. Dispersion and Classical Limit

Page map

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?

Common mistakes

Mistake	Correction
Treating a plane wave as an ordinary normalized particle state	use generalized normalization or form a normalizable packet
Identifying positive energy with rightward motion	remember the $k$ and $-k$ degeneracy
Calling phase velocity the particle velocity	track the packet envelope and its group velocity
Assuming a free packet keeps a rigid shape	inspect the curvature of the dispersion relation
Reading $\lvert\phi(p)\rvert^2$ as a position density	state the representation and integration variable
Confusing center motion with a complete classical limit	compare packet width with the resolved physical scales
Dropping the propagator prefactor	check the initial condition, units, and composition law
Assuming uncertainty saturation forbids later spreading	distinguish the lower bound from evolving correlations

Exercises

1. Energy degeneracy and current

For $\psi_k(x,t)=A e^{i(kx-\omega t)}$, find the energy and probability current. What changes under $k\mapsto-k$?

Solution

The free-particle dispersion relation gives

$$E = \hbar\omega = \frac{\hbar^2k^2}{2m}.$$

The one-dimensional current is

$$j = \frac{\hbar}{m} \operatorname{Im} \left( \psi_k^*\frac{\partial\psi_k}{\partial x} \right) = \frac{\hbar k}{m}\lvert A\rvert^2.$$

Changing $k$ to $-k$ leaves the energy unchanged and reverses the current.

2. Group and phase velocities

Derive $v_g$ and $v_{\mathrm{ph}}$ from $\omega(k)=\hbar k^2/(2m)$ and compare them with the classical velocity.

Solution

Differentiation gives

$$v_g = \frac{d\omega}{dk} = \frac{\hbar k}{m} = \frac{p}{m},$$

which is the classical free-particle velocity. Meanwhile,

$$v_{\mathrm{ph}} = \frac{\omega}{k} = \frac{\hbar k}{2m} = \frac12v_g.$$

The phase velocity tracks constant-phase surfaces, not the center of a localized packet.

3. Doubling the Gaussian width

At what time does the Gaussian uncertainty satisfy $\sigma_x(t)=\sqrt2\,\sigma_x$?

Solution

Using

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

the condition becomes

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

For positive time,

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

the characteristic spreading time.

4. Why two conditions are needed

Explain why $\Delta p\ll\lvert p_0\rvert$ alone does not guarantee particle-like classical motion.

Solution

The condition makes the relative velocity spread small, but the state may already be spatially broad compared with the apparatus or potential length scale. Classical particle-like motion also requires adequate localization and a time short enough that subsequent spreading remains unresolved. Momentum sharpness, spatial resolution, and observation time are separate conditions.

References

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