Wave Packet Derivation Today

: The width grows with time. Even in free space (no forces), a wave packet inevitably spreads because different ( k )-components have different phase velocities ( v_p = \omega/k = \hbar k/(2m) ). The initially synchronized components get out of phase.

Let’s simplify. Combine the exponentials:

is described by a plane wave. In one dimension, the wave function is: wave packet derivation

To create a localized disturbance, we must interfere multiple waves with different wave numbers ($k$) and frequencies ($\omega$). If we superimpose waves such that they interfere constructively in a small region of space and destructively everywhere else, we create a "packet" of energy.

Here’s a clear, step-by-step derivation of a from the superposition of plane waves, showing how it leads to a localized disturbance. : The width grows with time

Using the standard integral $\int_-\infty^\infty e^-ax^2 + bx dx = \sqrt\frac\pia e^b^2/4a$, we find: $$ \Psi(x,0) = \left( \frac12\pi\alpha^2 \right)^1/4 e^ik_0x e^-\fracx^24\alpha^2 $$

When you plug this into the integral and solve (using standard Gaussian integral techniques), the resulting wave function also takes a Gaussian shape in space. 4. Group vs. Phase Velocity Let’s simplify

When we derive the packet's movement, we find two distinct velocities: The speed of the individual ripples within the packet, Group Velocity (

vp=ωkv sub p equals the fraction with numerator omega and denominator k end-fraction Group Velocity (