@hokhani You're wrong when you claim in post #7 above that the group velocity is the average phase velocity. To resolve this, let us go over a concrete simple example when only two terms (with values ##k_1## and ##k_2##) are present in the summation in the ##\psi##-expansion. We will not use Taylor expansions explicitly, but perform simple algebraic manipulations.
First, when the amplitudes ##\alpha_k = \alpha(k)## in the expansion ##\psi(x,t) = \sum\limits_{k=k_1, k_2} \alpha(k) \exp\left(ikx - i\omega(k)t\right)## are essentially non-zero around some constant value ##k_0##, then we can replace them with a constant value ##\alpha_0 = \alpha(k_0)## so that both plane-wave components in our summation will have a common amplitude. In this way we will be able to write
$$
\psi(x,t) \approx \alpha_0 \left[ e^{ik_1x - i\omega_1t} + e^{ik_2x - i\omega_2t} \right] \rm{,}
$$
and then rewrite both exponents as
$$
e^{ik_1x - i\omega_1t} = e^{2\frac{ik_1x - i\omega_1t}{2}}
$$
(the same goes for the ##k_2##-dependent exponential term).
After this, we can take the common factor
$$
e^{\frac{ik_1x - i\omega_1t}{2}} e^{\frac{ik_2x - i\omega_2t}{2}}
$$
outside the parentheses, use the identity ##2\cos(\theta) = e^{i\theta} + e^{-i\theta}## and thus finally obtain that
$$
\psi(x,t) = 2\alpha_0 \exp\left(i \left[\frac{k_1+k_2}{2}\right] x - i \left[\frac{\omega_1+\omega_2}{2}\right] t\right) \cos\left( \left[\frac{k_2-k_1}{2}\right]x - \left[\frac{\omega_2-\omega_1}{2}\right]t \right) \rm{.}
$$
On the right hand-side of the above result we have a plane wave (i.e., this complex exponential term) with amplitude ##\sim \alpha_0##, wave vector ##\bar{k} = (k_1 + k_2)/2## and frequency ##\bar{\omega} = (\omega_1 + \omega_2)/2## -
here is where the average values appear. Then, just like in post #8 above, you can write down that for this plane wave the
phase velocity ##v_p = \bar{\omega}/\bar{k}## involves average quantities.
The remaining cosine term on the right hand-side above is the
envelope for the plane wave and we see that this envelope (which defines the overall "shape" of ##\psi##) involves the wave vector ##\kappa = (k_2-k_1)/2 = \Delta k / 2## and the frequency ##\Omega = (\omega_2 - \omega_1)/2 = \Delta\omega/2##, which are expressed as
differences of ##k##'s and ##\omega##'s, and
not their averages. If we now try to calculate the phase velocity just for this cosine term using the same prescription as was given above, then we will get that it is equal to ##\Omega / \kappa = \Delta\omega / \Delta k##. While formally this can be called the phase velocity of the envelope, it is a convention to call it a
group velocity (because the envelope groups together various waves, which then propagate "as a whole" inside the carrier envelope). Thus ##v_g = \Delta\omega / \Delta k##, and you can imagine that in the limit this becomes a derivative ##v_g = \partial \omega / \partial k##.
This example shows that, while there indeed appear some average values when considering this problem for ##\psi##-expansion, they nonetheless do not enter the group velocity - which is instead constructed using ratios of differences (which in the limit become derivatives).
This is correct, but it has nothing to do with taking average values, and therefore is a separate issue from the one raised in post #7 above. In other words, post #7 is wrong and post #9 is correct (but each post concern a different issue).
