PX156 - C3 - wave packets

idea

consider two plane waves:
$ψ_{1} = A \exp (i k x - i ω (- k) t)$
$ψ_{2} = A \exp - (i k x - i ω (- k) t)$
$ℏ ω (- k) = \frac{ℏ^{2} k^{2}}{2 m} = h ω (k) ⟹ ω (k) = ω (- k)$ $\begin{aligned} ψ & = ψ_{1} + ψ_{2} \\ = A \exp (- i ω (k) t) (\exp (i k x) + \exp (- i k x)) \\ = 2 A \exp (- i ω (k) t) \cos (k x) \end{aligned}$
- standing wave with amplitude, $2 A \cos (k x)$
$| ψ |^{2} = 4 A^{2} \cos^{2} (k x) = 2 A^{2} (1 + \cos (2 k x))$
consider two plane waves:
$ψ_{1} = A \exp (i k x - i ω (k_{1}) t)$
$ψ_{2} = A \exp - (i k x - i ω (k_{2}) t)$

$ψ = ψ_{1} + ψ_{2}$
- choosing $k_{1} = k_{0} + Δ k$ , $k_{2} = k_{0} - Δ k$ , $Δ k << k_{0}$
- expanding $ω (k_{1})$ and $ω (k_{2})$ as a taylor series:
  $ω (k_{1}) \approx ω (k_{0}) + Δ k \frac{d ω (k_{0})}{d k} + \dots = ω_{0} + Δ k ω_{0}^{'} + \dots$
  $ω (k_{2}) \approx ω (k_{0}) - Δ k \frac{d ω (k_{0})}{d k} + \dots = ω_{0} - Δ k ω_{0}^{'} + \dots$

\begin{aligned} ψ & = A \exp (i (k_{0} x - ω_{0} t + Δ k x - Δ k ω_{0}^{'} t)) + A \exp (i (k_{0} x - ω_{0} t - Δ k x + Δ k ω_{0}^{'} t)) \\ = A \exp (i (k_{0} x - ω_{0} t)) (\exp (i Δ k (x - ω_{0}^{'} t)) + \exp (- i Δ k (x - ω_{0}^{'} t))) \\ = 2 A \exp (i (k_{0} x - ω_{0} t)) \cos (Δ k (x - ω_{0}^{'} t)) \\ | ψ |^{2} & = 4 A^{2} \exp (i (k_{0} x - ω_{0} t)) (1 + \cos (2 Δ k (x - ω_{0}^{'} t))) \end{aligned}

this is a travelling wave, moving at speed, $v_{g} = ω_{0}^{'}$
- $\frac{d ω}{d k} = ω^{'} =$ group velocity
$ℏ ω (k) = \frac{ℏ^{2} k^{2}}{2 m}$ , and $ω^{'} = \frac{d}{d k} (\frac{ℏ k^{2}}{2 m}) = \frac{ℏ k}{m}$
$ω_{0}^{'} = \frac{ℏ k_{0}}{m} = \frac{p_{0}}{m}$

\int_{- \infty}^{+ \infty} | ψ |^{2} d x = \int_{- \infty}^{+ \infty} A^{2} d x = \infty \neq 1 u n l e s s A \equiv 0

consider a box of size $\pm L :$ $$\int_{-L}^{+L} A^{2},dx = 2LA^{2}= 1 \implies A= \frac{1}{\sqrt{2L}}$$

a particle as a wave packet

super position of many plane waves with wave numbers grouped around an average value, $k_{0} :$

ψ = \int d k a (k) \exp (i k x - i ω (k) t)

- where, $a (k)$ is the $k$ -dependent amplitude

a particle is represented by a combination that is grouped around a particular wavenumber, $k_{0}$ , with gaussian distribution of amplitudes of waves around $k_{0}$
$ω << k_{0} :$

ω (k) \approx ω (k_{0}) + ω_{0}^{'} (k - k_{0}) + \dots = ω_{0} + ω_{0}^{'} Δ k + \dots

for a wave packet where the distribution is narrow around $k_{0}$ , the wavefunction, $ψ$ , is given by the integral:

\begin{aligned} ψ & = \int d k a (k) \exp (i k_{0} x + i Δ k x - i ω_{0} t - i ω_{0}^{'} Δ k t) \\ = \exp (i k_{0} x - i ω_{0} t) \int d k a (k) \exp (i Δ k (x - ω_{0}^{'} t)) \end{aligned}

let $s = z - ω_{0}^{'} t : | ψ |^{2} = | f (s) |^{2}$
$ψ$ is a function in the form $f (s) = f (x - ω_{0}^{'} t)$ , which represents the travelling wave packet moving along at group velocity, $ω_{0}^{'}$
the shape of the wave packet is determined by the form of the distribution, $a (k)$