Group and Phase Velocity of a Localized Wave Packet



Group and Phase Velocity for a Localized Wave Packet

Build a Wave Packet

Consider a superposition of plane waves centered near k_0:

    \[ \Psi(x,t)=\int_{-\infty}^{\infty} A(k)\,e^{i\,[k x-\omega(k)t]}\,dk, \]

with A(k) sharply peaked at k_0. Expand the dispersion to second order:

    \[ \omega(k)\approx \omega_0+\omega'_0 (k-k_0)+\tfrac12 \omega''_0 (k-k_0)^2,\qquad \omega^{(n)}_0=\left.\frac{d^n\omega}{dk^n}\right|_{k_0}. \]

Factor out the carrier phase at (k_0,\omega_0):

    \[ \Psi(x,t)\approx e^{i(k_0 x-\omega_0 t)} \int A(k)\,\exp\Big\{i\big[(k-k_0)(x-\omega'_0 t)-\tfrac12 \omega''_0 (k-k_0)^2 t\big]\Big\}\,dk. \]

The envelope peaks where the linear phase vanishes, i.e., when x-\omega'_0 t=0.

Group Velocity

The envelope (group) propagates with

    \[ \boxed{\,v_g=\left.\frac{d\omega}{dk}\right|_{k_0}\,}. \]

The quadratic term \propto \omega''_0 governs spreading: if \omega''_0\neq 0, the packet broadens with time.

Phase Velocity

Surfaces of constant carrier phase satisfy k_0 x-\omega_0 t=\mathrm{const}, hence

    \[ \boxed{\,v_p=\frac{\omega_0}{k_0}\,}. \]

In general, therefore, v_p=\omega/k and v_g=d\omega/dk.

Important Special Cases

1) Free Nonrelativistic Particle (Schrödinger)

With de Broglie p=\hbar k, E=\hbar\omega, and E=\frac{p^2}{2m}:

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

Then

    \[ v_p=\frac{\omega}{k}=\frac{\hbar k}{2m}=\frac{v}{2},\qquad v_g=\frac{d\omega}{dk}=\frac{\hbar k}{m}=v, \]

so the group velocity equals the classical particle speed v; here \omega''(k)=\hbar/m\neq 0, so the packet spreads.

2) Electromagnetic Wave in Vacuum

With \omega=ck:

    \[ v_p=\frac{\omega}{k}=c,\qquad v_g=\frac{d\omega}{dk}=c, \]

and \omega''=0 → no spreading.

3) Relativistic Massive Particle

For E=\sqrt{p^2 c^2+m^2 c^4} and \omega=E/\hbar, k=p/\hbar:

    \[ \omega(k)=\frac{1}{\hbar}\sqrt{\hbar^2 k^2 c^2+m^2 c^4},\qquad v_p=\frac{\omega}{k}=\frac{E}{p}=\frac{c^2}{v},\qquad v_g=\frac{d\omega}{dk}=\frac{dE}{dp}=v, \]

giving the identity

    \[ \boxed{\,v_p\,v_g=c^2\,}\quad (m>0). \]

Here v_p>c is allowed (it carries no information); signals/energy follow v_g=v<c.

Summary:
Phase velocity tracks constant-phase surfaces: v_p=\omega/k.
Group velocity tracks the envelope (and typical information flow): v_g=d\omega/dk.
Dispersion (\omega''\neq 0) causes packet spreading; linear dispersion \omega\propto k avoids it.