<\body> The concept of Group velocity <\equation*> \(x,t)=dk \ |~>(k) e(k)t]> There is also a question about > factor. Here it is swallowed by |~>(k)>. (Shapes of pulses: broad frequency spectrum, sharp time; narrow frequency, long duration, close to monochromatic - or clean tone). Remember by flute and guitar (which is which?). Linear approximation <\equation*> \(k) \(k) = \(k)+|dk> (k-k) = \ \+|dk> (k-k) By simply setting t=0 <\equation*> \(x,0)=dk \ |~>(k) e This can easily be transformed to <\equation*> \ ex ] > >>dk \ |~>(k) e)x]> \ ex ] > >>dk> \ |~>(k>+k) e>x]> We can define the envelope (or modulation) as <\equation*> M(x)= ex ] >\(x,0) =>>dk> \ |~>(k>+k) e>x]> \; <\equation*> \ |~>(k)=>dx \(x,0) \ e Now we go back to <\equation*> \(x,t)=dk \ |~>(k) e(k)t]> and transform it like this <\equation*> dk \ |~>(k) ex]>e)x> et]> e|dk> (k-k)t]> after rearrangement <\equation*> \ ex] > \ et>dk \ |~>(k) e)x> e|dk> (k-k)t]> with anticipated change of variables <\equation*> \ (k-k)=k> We rewrite this as <\equation*> \ ex -\t] > \ dk \ |~>(k) e)x-|dk> (k-k)t]> and then obtain <\equation*> \ ex -\t] > \ dk> \ |~>(k+k) e>x-k>|dk>t]> or rearranged as <\equation*> \(x,t) \ \ \ ex -\t] >dk> \ |~>(k+k) \ e> [x-|dk>t]> With the definition of the envelope above <\equation*> M(x) =>>dk> \ |~>(k>+k) e>x]> the (x,t) >becomes <\equation*> \(x,t) \ \ \ ex -\t] >Mx-|dk>t \ \ We thus see that provided that the integral over > > \ is peaked close enough to k> so that the linear approximation holds, the wave propagates as a plane wave ex -\t] >>modulated by the envelope x-|dk>t \ \ > which moves with velocity =|dk> \ \ >. Note that if (k)=ck >, then the approximation is in fact an exact expression and the group velocity is the same as the phase velocity c. In general, the approximation does not display the fact that the envelope \ x)> \ >is in fact a function of time, x,t >>. Generally, a pulse broadens in the dispersive medium \; \; \; \; \; \; \; \; \; \; \; \;