, different components of the wave packet travel at different speeds, causing the packet to spread over time. 4. Group vs. Phase Velocity
: The center of the packet moves with the group velocity ( v_g = \fracd\omegadk\big|_k_0 = \frac\hbar k_0m ). This equals the classical particle velocity for a free particle. wave packet derivation
), leading directly to the . Dispersion: As time progresses, the quadratic term in the , different components of the wave packet travel
Key insights from this derivation:
ω(k)≈ω(k0)+dωdk|k0(k−k0)omega open paren k close paren is approximately equal to omega open paren k sub 0 close paren plus the fraction with numerator d omega and denominator d k end-fraction evaluated at k sub 0 end-evaluation open paren k minus k sub 0 close paren Phase Velocity : The center of the packet
Initially, consider a discrete sum of $N$ waves with closely spaced wave numbers: $$ \Psi(x,t) = \sum_n=0^N A_n e^i(k_n x - \omega_n t) $$
[ \Delta x(t) = \sqrt\alpha + \frac\hbar^2 t^24m^2 \alpha = \Delta x(0) \sqrt1 + \left( \frac\hbar t2m (\Delta x(0))^2 \right)^2 ]