Page 25 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

16 The wave function

1
Ø(x, t) ˆ Ø(î,0)ä(x ÿ ct ÿ î)dî ˆ Ø(x ÿ ct,0)
ÿ1
Thus, the wave packet Ø(x, t) has the same value at point x and time t that it
had at point x ÿ ct at time t ˆ 0. The wave packet has traveled with velocity c
without a change in its contour, i.e., it has traveled without dispersion. Since
the phase velocity v ph is given by ù 0 =k 0 ˆ c and the group velocity v g is given
by (dù=dk) 0 ˆ c, the two velocities are the same for an undispersed wave
packet.
We next consider the more general situation where the angular frequency
ù(k) is not proportional to jkj, but is instead expanded in the Taylor series
(1.13) about (k ÿ k 0 ). Now, however, we retain the quadratic term, but still
neglect the terms higher than quadratic, so that

ù(k) ù 0 ‡ v g (k ÿ k 0 ) ‡ ã(k ÿ k 0 ) 2
where equation (1.16) has been substituted for the ®rst-order derivative and ã
is an abbreviation for the second-order derivative

1 d ù
2
ã
2 dk 2 0
The phase in equation (1.27) then becomes
k(x ÿ î) ÿ ùt ˆ (k ÿ k 0 )(x ÿ î) ‡ k 0 (x ÿ î) ÿ ù 0 t

ÿ v g t(k ÿ k 0 ) ÿ ãt(k ÿ k 0 ) 2

ˆ k 0 x ÿ ù 0 t ÿ k 0 î ‡ (x ÿ v g t ÿ î)(k ÿ k 0 ) ÿ ãt(k ÿ k 0 ) 2
so that the wave packet (1.27) takes the form
1

e i(k 0 xÿù 0 t) 2
e
Ø ã (x, t) ˆ Ø(î, 0)e ÿik 0 î i(xÿv g tÿî)(kÿk 0 )ÿiãt(kÿk 0 ) dk dî
2ð
ÿ1
The subscript ã has been included in the notation Ø ã (x, t) in order to
distinguish that wave packet from the one in equations (1.14) and (1.15), where
the quadratic term in ù(k) is omitted. The integral over k may be evaluated
using equation (A.8), giving the result
1

e i(k 0 xÿù 0 t) ÿik 0 î ÿ(xÿv g tÿî) =4iãt
2
Ø ã (x, t) ˆ p  Ø(î, 0)e e dî (1:28)
2 iðãt
ÿ1
Equation (1.28) relates the wave packet at time t to the wave packet at time
2
t ˆ 0 if the k-dependence of the angular frequency includes terms up to k .
The pro®le of the wave packet Ø ã (x, t) changes as time progresses because of