# Where do the group and phase velocity terms come from in the wave function of the propagation of a dispersive wave packet?

I am reading Zettili's QM concepts and applications (second edition), and in section 1.8.3 where he discusses the motion of wave packets, we consider a wave packet where angular frequency $$\omega$$ is a function of the wave number $$k$$: $$\omega = \omega(k)$$.

We assume that $$\phi(k)$$ is a function $$g(k-k_0)$$ that is narrow and peaks at $$k=k_0$$, so we can Taylor expand $$\omega(k)$$ about $$k_0$$ as: $$\omega(k) = \omega(k_0)+(k-k_0)\left.\frac{d\omega(k)}{dk}\right|_{k=k_0} + \frac{1}{2}(k-k_0)^2\left.\frac{d^2\omega(k)}{dk^2}\right|_{k=k_0} + ...$$ $$\omega(k) = \omega(k_0)+(k-k_0)v_g + (k-k_0)^2\alpha + ...$$

then all we would need to do is substitute $$\omega(k)$$ into the equation $$\Psi(x,t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty}\phi(k)e^{i(kx-\omega t)}dk$$ . In the book, this is shown as: $$\Psi(x,t) = \frac{1}{\sqrt{2\pi}}e^{ik_0(x-v_{ph}t)} \int_{-\infty}^{+\infty} g(k-k_0)e^{i(k-k_0)(x-v_gt)} e^{-i(k-k_0)^2 \alpha t + ...} dk$$

I don't understand where the phase velocity came from at all, and how did he get $$e^{i(k-k_0)(x-v_gt)}$$? Shouldn't it simply be $$e^{\omega(k_0)-i(k-k_0)v_gt}$$?

Also a sub question: In the integral, $$e^{i(k-k_0)(x-v_gt)} e^{-i(k-k_0)^2 \alpha t + ...}$$ ends with a $$+...$$, but if you write out more terms, they would all start with a minus sign because of the minus sign in $$-\omega t$$, right?

Zitteli is simply writing the exponent as $$\begin{eqnarray} i(kx-\omega(k) t) &=& i(kx-[\omega(k_0)+(k-k_0)v_g +(k-k_0)^2\alpha + ...]t) \end{eqnarray}$$ and then adding and subtracting $$k_0 x$$, he writes $$$$-\omega(k_0) t = k_0\left (x-\frac{\omega(k_0)}{k_0} t\right ) - k_0 x \,.$$$$ The phase velocity is defined to be $$v_{ph} = \frac{\omega(k_0)}{k_0}$$, so this is $$$$-\omega(k_0) t = k_0(x-v_{ph} t) - k_0 x \,.$$$$ Substituting gives Zettili's expression.
Often $$+...$$ just means there are more terms that are ignored and doesn't imply a sign (over even a phase).