1.11: Examples of wavepackets

A typical Gaussian wavepacket is shown in Figure 1.11.1 in both its real space and k-space representations. Initially the probability distribution is centered at x = 0 and k = 0. If we shift the wavepacket in k-space to an average value \(\langle k\rangle = k_{0}\), this is equivalent to multiplying by a phase factor \(\text{ exp}[ik_{0}x]\) in real space. Similarly, shifting the center of the wavepacket in real space to \(\langle x\rangle = x_{0}\) is equivalent to multiplying the k-space representation by a phase factor \(\text{ exp}[ikx_{0}]\).

Table \(\PageIndex{1}\): A summary of shift transformations in real and inverse coordinates.

Real coordinates (x,t)	\(\rightleftharpoons\)	Inverse coordinates (k,\(\omega\))
shift by \(x_{0}\)	\(\rightleftharpoons\)	\(\times \text{ exp}[-ikx_{0}]\)
\(\times \text{ exp}[ik_{0}x]\)	\(\rightleftharpoons\)	shift by \(k_{0}\)
shift by \(t_{0}\)	\(\rightleftharpoons\)	\(\times \text{ exp}[i\omega t_{0}]\)
\(\times \text{ exp}[-i\omega_{0}t]\)	\(\rightleftharpoons\)	shift by \(\omega_{0}\)

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