Group vs Phase Velocity

Animation

Theoretical Context

In quantum mechanics and wave mechanics, there is a difference between the so-called phase velocity and the group velocity. The notion of these concepts arises when considering the superposition of multiple waves. This superposition can result in the combination of an ‘envelope’ wave and a ‘carrier’ wave.

To illustrate the concept of phase and group velocities, we consider the superposition of two plane waves. We consider two waves \Psi_{1,2} of the form:

\begin{equation*}
\Psi_1 = A\cos(\omega_1t-k_1x),\quad\quad\quad \Psi_2 = A\cos(\omega_2t-k_2x),
\end{equation*}

where we took the amplitude of the waves to be identical (A_1=A_2\equiv A), \omega is the angular frequency and k the wavenumber. From now on, we take A\equiv 1 for simplicity.

Now consider the superposition of these two waves, i.e. \Psi_{1+2}=\Psi_1+\Psi_2. Using the sum-to-product trig identity:

\cos\theta_1+\cos\theta_2=2\cos\Big(\frac{\theta_1-\theta_2}{2}\Big)\cos\Big(\frac{\theta_1+\theta_2}{2}\Big),

we can write the superposition of the plane waves as:

\begin{equation*}
\Psi_{1+2}=2\cos\Big(\frac{\Delta \omega}{2}t-\frac{\Delta k}{2}x\Big)\cos(\bar{k}x-\bar{\omega}t). 
\end{equation*}

Above, we have defined \Delta\omega = \omega_1-\omega_2 and the average frequency is defined as \bar{k}=(k_1+k_2)/2, the same definitions hold for the wavenumber. Note that the superposition can be written as the product of an envelope and carrier wave \Psi_{1+2}\equiv\Psi_\mathrm{env}\Psi_\mathrm{car}, where:

\begin{equation*}
\Psi_\mathrm{env}\equiv 2\cos\Big(\frac{\Delta \omega}{2}t-\frac{\Delta k}{2}x\Big),\quad\quad\quad \Psi_\mathrm{car}\equiv \cos(\bar{k}x-\bar{\omega}t).
\end{equation*}

The factorization has a simple interpretation, the fast oscillating carrier wave is modulated by (or contained in) the envelope. The carrier moves at the phase velocity:

v_p\equiv \frac{\bar{\omega}}{\bar{k}}=\frac{\omega(k)}{k},

where the final form is the continuous generalization of the first. The envelope moves at the group velocity:

v_g\equiv \frac{\Delta \omega}{\Delta k}=\frac{d\omega}{dk},

where the second equality defines the continuous generalization of the first definition. For the continuous definition to be useful, an explicit expression for \omega(k), known as the dispersion relation, is required.

Physical Interpretation

We will consider two types of dispersion: non-relativistic and relativistic. Non-relativistic dispersion corresponds to \omega(k)\propto k^2 and relativistic dispersion corresponds to \omega(k)\propto k. Non-relativistic dispersion occurs for instance for a free quantum particle. Its total energy can be written as E=\hbar^2k^2/2m. Using the definition E=\hbar\omega, we then have a quadratic dispersion relation:

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

Photons are relativistic and therefore satisfy a relativistic dispersion relation. Since they are massless, their energy is E=pc and their momentum is p=\hbar k, so that the dispersion relation becomes linear:

\omega(k) =ck.

Note that linear and quadratic dispersion relations always satisfy v_g=v_p and v_g=2v_p, respectively. This is indeed confirmed by the two animations.

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