PX156 - C5 - top-hat distribution

a (k) = {\begin{cases} A & f o r & k_{0} - \frac{w}{2} < k < k_{0} + \frac{w}{2} \\ 0 & o t h e r w i s e \end{cases}

\begin{aligned} f (s) & = A \int_{- \frac{w}{2}}^{\frac{w}{2}} d k^{'} \exp (i k^{'} s) \\ = A {[\frac{\exp (i k^{'} s)}{i s}]}_{- \frac{w}{2}}^{\frac{w}{2}} \\ = \frac{2 A}{s} \sin (\frac{w s}{2}) \\ ∴ | f (s) |^{2} & = \frac{4 A^{2}}{s^{2}} \sin^{2} (\frac{w s}{2}) \end{aligned}

the width in $k$ is $w$
the width in $s$ is inversely proportional to $w$
important result: a wave packet with width, $w$ , can represent a particle localised in a region of width of order $\frac{1}{w}$