PX156 - C4 - gaussian distribution

a (k) = A \exp (- \frac{(k - k_{0})^{2}}{2 σ^{2}})

where, $σ =$ standard deviation

\begin{aligned} f (s) = f (x - v_{g} t) & = A \int_{- \infty}^{\infty} d k^{'} \exp (- \frac{k^{' 2}}{2 σ^{2}}) \exp (i k^{'} s) \\ = A \int_{- \infty}^{\infty} d k^{'} \exp (- \frac{1}{2} {(\frac{k^{'}}{σ} - i s σ)}^{2} - \frac{s^{2} σ^{2}}{2}) \\ = A \exp (- \frac{s^{2} σ^{2}}{2}) σ \int_{- \infty}^{\infty} \frac{d k^{'}}{σ} \exp (- \frac{1}{2} {(\frac{k^{'}}{σ} - i s σ)}^{2}) \\ = A \exp (- \frac{s^{2} σ^{2}}{2}) σ \sqrt{2 π} \\ ∴ | f (s) |^{2} & = 2 A^{2} σ^{2} π \exp (- s^{2} σ^{2}) \end{aligned}

- also a gaussian

w^{2} = \frac{\int | a (k) |^{2} (k - k_{0})^{2} d k}{\int d k | a (k) |^{2}}

\begin{aligned} \frac{1}{2} A & = A \exp (- \frac{Δ k^{2}}{2 σ^{2}}) \\ - \ln 2 & = - \frac{Δ k^{2}}{2 σ^{2}} \\ Δ k^{2} & = 2 σ^{2} \ln 2, σ \sqrt{2 \ln 2} \\ w & = 2 σ \sqrt{2 \ln 2} \approx 23556 \end{aligned})

w^{'} = 2 Δ s = 2 \frac{\sqrt{\ln 2}}{σ} \sim \frac{1}{w}

\begin{aligned} ω & = \frac{ℏ k^{2}}{2 m} \\ \frac{d ω}{d k} & = \frac{ℏ k}{m} \\ v_{g} & = \frac{d}{d k} ω (k_{0}) \\ = \frac{ℏ k}{m} \\ = \frac{p}{m} \end{aligned}