Wave Packets & Dispersion

A wave packet is a spatially localised disturbance formed by superposing many sinusoidal waves with a range of wave numbers $k$ centred around $k_0$. The result is an envelope (the packet's outline) that travels at the group velocity, while individual wave crests inside travel at the phase velocity.

In a non-dispersive medium (e.g., light in vacuum), all frequency components travel at the same speed and the packet maintains its shape. In a dispersive medium (e.g., glass, a quantum potential), different $k$ components travel at different speeds, causing the packet to spread over time.

This concept bridges classical waves and quantum mechanics: in quantum mechanics, a free particle is described by a wave packet; its position is the centre of the packet and its momentum is $\hbar k_0$. The inevitable spreading of the packet reflects the Heisenberg uncertainty principle.

Phase velocity (speed of individual crests):

$v_p = \frac{\omega}{k}$

Group velocity (speed of the envelope/packet):

$v_g = \frac{d\omega}{dk}\bigg|_{k=k_0}$

Heisenberg uncertainty principle (position-momentum form):

$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}, \quad \text{since } p = \hbar k$

A narrower packet in position space ($\Delta x$ small) requires a wider spread in $k$-space ($\Delta k$ large), so $\Delta x \cdot \Delta k \geq \frac{1}{2}$.

For matter waves, a free electron has dispersion relation $\omega = \hbar k^2 / 2m$.

Phase velocity: $v_p = \omega/k = \hbar k / 2m$

Group velocity: $v_g = d\omega/dk = \hbar k / m = p/m$ — exactly the classical particle velocity.

For a packet with $\Delta x = 1\,\text{nm}$ (atomic scale):

$\Delta k \geq \frac{1}{2\Delta x} = \frac{1}{2 \times 10^{-9}} = 5\times10^8\,\text{m}^{-1}$

$\Delta p = \hbar \Delta k \geq (1.055\times10^{-34})(5\times10^8) \approx 5.3\times10^{-26}\,\text{kg m/s}$

This sets a hard lower limit on momentum uncertainty for an electron confined to 1 nm.

Laser pulses in optical fibres are wave packets. Dispersion in the fibre causes different frequency components to arrive at different times, spreading the pulse and limiting data rates — engineers use dispersion-shifted fibres and chirped pulse amplification to compensate. In quantum computing, controlling the spread and coherence of electron wave packets is essential for maintaining qubit fidelity.

1. A wave packet has dispersion relation $\omega = ck$ (light in vacuum). What are $v_p$ and $v_g$? Does the packet spread?

2. An electron is confined to a box of width $\Delta x = 0.1\,\text{nm}$. Estimate the minimum uncertainty in its momentum $\Delta p$ using $\Delta x \cdot \Delta p \geq \hbar/2$.