Editing Group velocity (section)

===Derivation===
One derivation of the formula for group velocity is as follows.<ref name=Griffiths>
{{cite book
 | author=Griffiths, David J.
 | title=Introduction to Quantum Mechanics
 | url=https://archive.org/details/introductiontoqu00grif_200
 | url-access=limited
 | publisher=[[Prentice Hall]]
 | year=1995
 | page=[https://archive.org/details/introductiontoqu00grif_200/page/n61 48]
| isbn=9780131244054
 }}</ref><ref>
{{cite book
 | title = Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers
 | edition = 2nd
 | author = David K. Ferry
 | publisher = CRC Press
 | year = 2001
 | isbn = 978-0-7503-0725-3
 | pages = 18–19
 | url = https://books.google.com/books?id=imvYBULWPMQC&pg=PA18
| bibcode = 2001qmid.book.....F
 }}</ref>

Consider a [[wave packet]] as a function of position {{math|''x''}} and time {{math|''t'': ''α''(''x'',''t'')}}.

Let {{math|''A''(''k'')}} be its [[Fourier transform]] at time {{nowrap|{{math|''t'' {{=}} 0}}}},
:<math> \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}.</math>

By the [[superposition principle]], the wavepacket at any time  {{math|''t''}} is
:<math> \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)},</math>
where  {{math|''ω''}} is implicitly a function of  {{math|''k''}}.

Assume that the wave packet  {{math|''α''}} is almost [[monochromatic]], so that  {{math|''A''(''k'')}} is sharply peaked around a central [[wavenumber]]  {{math|''k''<sub>0</sub>}}.

Then, [[linearization]] gives
:<math>\omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0</math>
where 
:<math>\omega_0 = \omega(k_0)</math> and <math>\omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0}</math> 
(see next section for discussion of this step). Then, after some algebra,
:<math> \alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}.</math>

There are two factors in this expression. The first factor, <math>e^{i\left(k_0 x - \omega_0 t\right)}</math>, describes a perfect monochromatic wave with wavevector  {{math|''k''<sub>0</sub>}}, with peaks and troughs moving at the [[phase velocity]] <math>\omega_0/k_0</math>  within the envelope of the wavepacket.

The other factor, 
:<math>\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}</math>, 
gives the envelope of the wavepacket. This envelope function depends on position and time ''only'' through the combination <math>(x - \omega'_0 t)</math>.

Therefore, the envelope of the wavepacket travels at velocity 
:<math>\omega'_0 = \left.\frac{d\omega}{dk}\right|_{k=k_0}~,</math> 
which explains the group velocity formula.