In some texts of fundamental quantum mechanics, it introduces the wave packet by Fourier transformation of a momentum wave into a spatial version. This is easy to understand because, analogy to the optical wave, a typical beam could compose waves of more than one frequencies. The general form is something like this(adsbygoogle = window.adsbygoogle || []).push({});

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\psi(x) = \dfrac{1}{\sqrt{2\pi}} \int dk \psi(k) \exp(ikx)

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This is quite straightforward. However, I have hard time to understand why the time evolution of a general wave packet is of the following form involving ##\omega(k)t##

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\psi(x, t) = \dfrac{1}{\sqrt{2\pi}} \int dk \psi(k) \exp[ikx - i\omega(k)t]

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In all texts I have in hand, above expression is given directly. From math context, it seems that it is trying to tackle the time aspect of ##k## with ##\omega(k)t## but it of above form instead of the following?

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\psi(x, t) = \dfrac{1}{\sqrt{2\pi}} \int dk \psi(k) \exp[ik(t)x]

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I am looking for more physical intuitive explanation of the term ##i\omega(k)t##

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# I About the time evolution of a wave packet

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