Wave Packets in Early Wave Mechanics

Matter waves created an immediate interpretive problem. A single de Broglie wave with definite momentum is spread over all space, but experiments prepare beams, tracks, spots, and localized detections. Wave packets were the bridge: localized quantum states can be built by superposing many plane-wave components with nearby wavelengths.

The de Broglie relation fixes each component’s wavelength. Fourier superposition explains how those components combine into a localized state; free-particle dynamics predicts how that state moves.

Plane Waves and the Localization Problem

The de Broglie relation assigns a wavelength to a momentum:

$$\lambda=\frac{h}{p}, \qquad p=\hbar k.$$

A plane-wave component has the schematic form

$$\psi_k(x,t) = A e^{i(kx-\omega t)}.$$

It is useful because it has definite wave number and definite momentum. But it does not describe a localized particle on the full line. Its probability density is constant:

$$\lvert\psi_k(x,t)\rvert^2 = \lvert A\rvert^2.$$

There is no packet center, no finite spatial width, and no localized bump to track. This was not a small technical inconvenience. If matter waves were to describe electrons arriving at finite regions of detectors, the theory needed localized states.

Superposition of Waves

A wave packet solves the localization problem by superposing many wave components:

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

The function $a(k)$ gives the amplitude for different wave numbers. If $a(k)$ is concentrated near $k_0$, the packet has a fairly well-defined mean momentum. A broad $a(k)$ permits sharper spatial localization, but its phases also matter; the width of $|a(k)|^2$ alone does not determine the packet’s spatial width. The tradeoff is the physical content behind the later Position-Momentum Uncertainty relation.

An unchirped Gaussian packet and its wave-number density illustrate $\sigma_x\sigma_k=1/2$. The oscillations are the real part of the complex amplitude; the dashed curves show its magnitude, not the probability density. Position and wave-number axes use reciprocal units. Narrower position distributions require broader momentum distributions.

This is the key conceptual move. The wave nature of matter is not represented by one infinite sinusoid. Physical localized states are superpositions. The individual wave components are mathematical building blocks; the packet is the normalizable state that can approximate a localized particle.

Group Velocity

The packet center is governed by the dispersion relation $\omega(k)$. If $a(k)$ is sharply peaked near $k_0$, expand $\omega(k)$:

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

Keeping this linear approximation, the packet envelope moves with group velocity

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

For a nonrelativistic free particle,

$$E=\frac{p^2}{2m}, \qquad p=\hbar k, \qquad E=\hbar\omega,$$

so

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

The group velocity is therefore

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

That is exactly the classical velocity of a free particle with momentum $p_0$. This agreement was one reason wave packets made matter waves physically intelligible: a localized envelope could move like a particle even though it was built from waves.

The phase velocity of an individual component is different:

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

This phase velocity uses the nonrelativistic energy zero with the rest energy subtracted. Adding a constant to all energies changes $\omega/k$ but not $d\omega/dk$ or any probability. Using the relativistic total energy instead gives $v_{\mathrm{ph}}=c^2/v$ and $v_g=v$. The superluminal phase velocity carries no signal.

Wave Packet Spreading

The same dispersion relation that gives the group velocity also causes spreading. A better expansion is

$$\omega(k) = \omega(k_0) +\omega'(k_0)(k-k_0) +\frac{1}{2}\omega''(k_0)(k-k_0)^2+\cdots.$$

The constant term gives an overall phase. The linear term translates the packet center. The quadratic term changes the relative phases of nearby wave-number components and therefore changes the shape of the envelope.

For a nonrelativistic free particle,

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

so a free packet generally changes width and eventually spreads. Position–momentum correlations can make it contract initially; dispersion does not imply monotonic broadening for every preparation. For an initially unchirped minimum-uncertainty Gaussian with position standard deviation $\sigma_0$,

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

Here “unchirped” means no initial quadratic position-dependent phase. The density broadens while its integral remains one; spreading does not mean probability is lost.

The detailed spreading formulas live in Wave Packet Spreading and Gaussian Wave Packets. The historical lesson is simpler: a wave account of matter can recover particle-like motion only as an approximation, and the limits of that approximation are physically meaningful.

Why a Packet Is Not a Permanent Corpuscle

A localized envelope helps explain approximate particle motion, but a free Schrödinger packet generally changes shape. That prevented the simple identification of an electron with a rigid classical wave lump. A packet is a state with position and momentum distributions; localized detection requires the probability rule as well as the wave equation.

Common Mistakes

• Treating a plane wave as a localized particle. A plane wave is a momentum eigenstate idealization, not a normalizable localized state.
• Saying a packet is a small classical wave of matter. The packet is a quantum state whose squared magnitude gives probabilities.
• Identifying phase velocity with particle velocity. The packet envelope moves with group velocity.
• Forgetting that localization requires momentum spread.
• Treating packet spreading as a failure of probability conservation.
• Assuming a packet always moves like a rigid classical object. That is an approximation controlled by dispersion, mass, packet width, and time scale.

Exercises

1. Why can a single full-line plane wave not represent a localized particle?

Solution

A plane wave has the form $Ae^{i(kx-\omega t)}$, so its probability density is

$$\lvert Ae^{i(kx-\omega t)}\rvert^2 = \lvert A\rvert^2.$$

That density is constant in space. There is no finite-width region where the particle is localized, and the state is not square-normalizable on the full line.

2. Derive the group velocity for a nonrelativistic free-particle packet centered at $k_0$.

Solution

For a nonrelativistic free particle,

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

Therefore

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

Since $p_0=\hbar k_0$, this gives

$$v_g=\frac{p_0}{m},$$

the classical free-particle velocity.

3. Explain qualitatively why a sharply localized packet tends to spread rapidly.

Solution

Sharp localization requires many Fourier components, which means a broad momentum distribution. For a free nonrelativistic particle, different momentum components have different velocities $p/m$. Those components separate over time, so the spatial probability distribution broadens.

4. In what sense do wave packets reconcile particle-like motion with wave-like diffraction?

Solution

The components of the packet carry phases and wavelengths, so the state can diffract and interfere. The superposed envelope can still be localized enough to move with an approximate center and group velocity. Thus the packet gives particle-like motion as an approximation built from wave-like amplitudes, not as a return to classical particles.

5. An electron starts in the Gaussian state described above with $\sigma_0=1.0\,\mathrm{nm}$. When does its width reach $\sqrt2\,\sigma_0$? Use $m_e=9.11\times10^{-31}\,\mathrm{kg}$ and $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$.

Solution

Set $\hbar t/(2m_e\sigma_0^2)=1$. Then $t=2m_e\sigma_0^2/\hbar\simeq1.73\times10^{-14}\,\mathrm s$. Doubling the initial width multiplies this time by four; increasing the mass also slows spreading. A narrow free electron packet is not a rigid object.

References

• M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 37, 863-867, 1926, DOI: 10.1007/BF01397477.
• L. de Broglie, “Recherches sur la théorie des quanta,” Annales de Physique, 10th series, 3, 22–128, 1925, DOI: 10.1051/anphys/192510030022.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• E. Schrödinger, “Quantisierung als Eigenwertproblem,” Annalen der Physik 79, 361-376, 1926, DOI: 10.1002/andp.19263840404.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.