Definition

Applying Eq.(2.67) to Eq.(2.68) one finds that the typical wave packet is

$$P^\varepsilon_{j\ell}(t) = \int^{(j+1)\varepsilon}_{j\varepsilon} \!\!\!\!\!\! d\omega \unde... ...c{2\pi\ell}{\varepsilon}) j \varepsilon} } {i(t-\frac{2\pi\ell}{\varepsilon}) }$$

$$~ \begin{array}{c} \textrm{unitary}\\ \textrm{xformation} \end... ...~ \begin{array}{c} \textrm{wave}\\ \textrm{packet} \end{array}$$

$$= \frac{2}{\sqrt{2\pi\varepsilon}} \frac{\displaystyle \sin \left[ ... ...isplaystyle i(t-\frac{2\pi\ell}{\varepsilon})(j+\frac{1}{2})\varepsilon} \} ~~.$$ (269)

Notation: For the purpose of notational efficiency we shall suppress the superscript $\varepsilon$ in the wavepacket name $P^\varepsilon_{j\ell}(t)$ throughout the remainder of Section 2.4.2. Thus we use simply $P_{j\ell }(t)$ instead. However, in the upcoming Sections 2.5 and 2.6 we shall always highlight $\varepsilon$ by explicitly writing $P^\varepsilon_{j\ell}(t)$ .