Page 26 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.3 Dispersion of a wave packet 17

the factor t ÿ1=2 before the integral and the t in the exponent within the integral.
If we select a speci®c form for the wave packet at time t ˆ 0, the nature of this
time dependence becomes more evident.

Gaussian wave packet
Let us suppose that Ø(x, 0) has the gaussian distribution (1.20) as its pro®le, so
that equation (1.14) at time t ˆ 0is

2 2
e
Ø(î,0) ˆ e ik 0 î B(î,0) ˆ p 1  e ik 0 î ÿá î =2 (1:29)
2ð
Substitution of equation (1.29) into (1.28) gives
e i(k 0 xÿù 0 t) 1 ÿá î =2 ÿ(xÿv g tÿî) =4iãt
2 2
2
Ø ã (x, t) ˆ p  e e dî
2ð 2iãt
ÿ1
The integral may be evaluated using equation (A.8) accompanied with some
tedious, but straightforward algebraic manipulations, yielding
e i(k 0 xÿù 0 t) 2 2 2
 e ÿá (xÿv g t) =2(1‡2iá ãt) (1:30)
Ø ã (x, t) ˆ p
2
2ð(1 ‡ 2iá ãt)
The wave packet, then, consists of the plane wave exp i[k 0 x ÿ ù 0 t] with its
amplitude modulated by
1 2 2 2
 e ÿá (xÿv g t) =2(1‡2iá ãt)
p
2
2ð(1 ‡ 2iá ãt)
which is a complex function that depends on the time t. When ã equals zero so
that the quadratic term in ù(k) is neglected, this complex modulating function
reduces to B(x, t) in equation (1.20). The absolute value jØ ã (x, t)j of the wave
packet (1.30) is given by
1 2 2 4 2 2
jØ ã (x, t)jˆ e ÿá (xÿv g t) =2(1‡4á ã t ) (1:31)
4 2 2 1=4
(2ð) 1=2 (1 ‡ 4á ã t )
We now contrast the behavior of the wave packet in equation (1.31) with that
of the wave packet in (1.20). At any time t, the maximum amplitudes of both
occur at x ˆ v g t and travel in the positive x-direction with the same group
velocity v g . However, at that time t, the value of jØ ã (x, t)j is 1=e of its
maximum value when the exponent in equation (1.31) is unity, so that the half
width or uncertainty Äx for jØ ã (x, t)j is given by
p 
2 p 
4 2 2
Äx ˆjx ÿ v g tjˆ 1 ‡ 4á ã t
á
Moreover, the maximum amplitude for jØ ã (x, t)j at time t is given by
4 2 2 ÿ1=4
(2ð) ÿ1=2 (1 ‡ 4á ã t )