Wave Packets1
Page 62
Definition
Assert that
  k0 , x0  

 f  k  k  exp ik  x  x   dk (1.1)
0
0

Let y=k-k0
  k0 , x0  

 f  y  exp i  y  k  x  x   dy (1.2)
0
0

Evaluate the integral in accordance with ..\..\definitions\AiGau\GuidingFunction.docx
So that
  k0 , x0  

20
20
 exp i  t
i 1
i
 ko  x  x0  
Is a packet at x0.
Group velocity
Uses the fact that in free space w = ck
E  x, t  

 f  k  k  exp ik  x  x   iwt  dk
0
0



 f  k  k  exp ik  x  x
0
0
 ct   dk
(2.1)

Suppose w is a function of k
E  x, t  

 f  k  k  exp ik  x  x   iw  k  t  dk (2.2)
0
0

The maximum of the packet occurs when the exponential has an extremum.
This occurs when
1
David Bohm, Quantum Theory, Prentice Hall (1951) pp 60-70
  ik  x  x0   iw  k  t 
k
(2.3)
or
x  x0  t
0
w  k 
k
This is the velocity of the maximum of the wave packet.