The Speed of a Wavepacket
A quantum particle is a Wavepacket—a sum of many individual waves. We must distinguish between two speeds:
- Phase Velocity (\(v_p = \omega/k\)): The speed of the individual ripples.
- Group Velocity (\(v_g = d\omega/dk\)): The speed of the "envelope" or the packet itself.
The group velocity is the speed that we actually measure as the particle's velocity.
Worked Examples
Example 1: Measuring the Particle
For matter waves, the phase velocity is actually half of the measured velocity, and it can even be faster than the speed of light! However, information and mass only travel at the Group Velocity, which always matches the classical velocity \(v = p/m\) and is always slower than light.
The Bridge to Quantum Mechanics
The difference between phase and group velocity explains Dispersion. As a quantum wavepacket travels, it "spreads out" because its individual frequency components move at different speeds. This is why a perfectly localized electron doesn't stay localized for long—it naturally diffuses into a wider cloud of probability over time. Controlling this dispersion is the main challenge in Quantum Communication, where we try to send quantum bits (qubits) over long distances without them "blurring" away.