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 \(\text{ exp}[ikx_{0}]\).
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 \(\omega_{0}\) |