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1.11: Examples of wavepackets

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    49374
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    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 \(\omega_{0}\)
    Screenshot 2021-04-14 at 10.58.23.png
    Figure \(\PageIndex{1}\): Three Gaussian wavepackets. In (a) the average position and wavenumber of the packet is x=0 and k=0, respectively. In (b) the average position has been shifted to \(\langle x\rangle =x_{0}\). In (c) the average wavenumber has been shifted to \(\langle k\rangle =k_{0}\).

    This page titled 1.11: Examples of wavepackets is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.