Page 23 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 23
14 The wave function
is a property of Fourier transforms. In order to localize a wave packet so that
the uncertainty Äx is very small, it is necessary to employ a broad spectrum of
plane waves in equations (1.11) or (1.17). The function A(k) must have a wide
distribution of wave numbers, giving a large uncertainty Äk. If the distribution
A(k) is very narrow, so that the uncertainty Äk is small, then the wave packet
becomes broad and the uncertainty Äx is large.
Thus, for all wave packets the product of the two uncertainties has a lower
bound of order unity
ÄxÄk > 1 (1:23)
The lower bound applies when the narrowest possible range Äk of values for k
is used in the construction of the wave packet, so that the quadratic and higher-
order terms in equation (1.13) can be neglected. If a broader range of k is
allowed, then the product ÄxÄk can be made arbitrarily large, making the
right-hand side of equation (1.23) a lower bound. The actual value of the lower
bound depends on how the uncertainties are de®ned. Equation (1.23) is known
as the uncertainty relation.
A similar uncertainty relation applies to the variables t and ù. To show this
relation, we write the wave packet (1.11) in the form of equation (B.21)
1 1 i(kxÿùt)
Ø(x, t) p G(ù)e dù (1:24)
2ð ÿ1
where the weighting factor G(ù) has the form of equation (B.22)
1 1 ÿi(kxÿùt)
G(ù) p Ø(x, t)e dt
2ð ÿ1
In the evaluation of the integral in equation (1.24), the wave number k is
regarded as a function of the angular frequency ù, so that in place of (1.13) we
have
dk
k(ù) k 0 (ù ÿ ù 0 )
dù 0
If we neglect the quadratic and higher-order terms in this expansion, then
equation (1.24) becomes
Ø(x, t) C(x, t)e i(k 0 xÿù 0 t)
where
1 1
C(x, t) p A(ù)e ÿi[tÿ(dk=dù) 0 x](ùÿù 0 ) dù
2ð ÿ1
As before, the wave packet is a plane wave of wave number k 0 and angular
frequency ù 0 with its amplitude modulated by a factor that moves in the
positive x-direction with group velocity v g , given by equation (1.16). Following
