Introduction: Two Speeds
Now we understand why this matters for quantum mechanics. The phase velocity describes the ripples; the group velocity describes the particle. For matter waves, these are different, with profound consequences.
Definitions
Phase velocity: \(v_p = \frac{\omega}{k}\) (speed of wavefronts)
Group velocity: \(v_g = \frac{d\omega}{dk}\) (speed of wave packet envelope)
For Free Particle
Dispersion relation: \(\omega = \frac{\hbar k^2}{2m}\)
Phase velocity: \(v_p = \frac{\omega}{k} = \frac{\hbar k}{2m} = \frac{p}{2m}\)
Group velocity: \(v_g = \frac{d\omega}{dk} = \frac{\hbar k}{m} = \frac{p}{m} = v_\text{classical}\)
The Key Result
The group velocity equals the classical particle velocity! The wave packet moves like a classical particle, even though individual wavefronts move at half that speed.
\[v_g = 2v_p\]Worked Example: Relativistic Case
For relativistic particles, \(E^2 = p^2c^2 + m^2c^4\), and:
\[v_p = \frac{E}{p} = \frac{c}{\sqrt{1-(mc/p)^2}} > c\] \[v_g = \frac{dE}{dp} = \frac{pc^2}{E} < c\]Phase velocity exceeds \(c\), but group velocity (the physical velocity) never does.
The Quantum Connection
This resolves an apparent paradox: phase velocity can exceed \(c\), but no information travels faster than light because particles and energy travel at the group velocity. The distinction between phase and group velocity is essential for understanding wave packet dynamics and quantum tunneling times.