Page 30 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.5 Heisenberg uncertainty principle 21

Thus, the velocity v of the particle is associated with the group velocity v g of
the wave packet

v ˆ v g
If the constant potential energy V in equation (1.35) is set at some arbitrary
value other then zero, then equation (1.39) takes the form
"k 2 V
ù(k) ˆ ‡
2m "
and the phase velocity v ph becomes
"k 0 V
v ph ˆ ‡
2m "k 0
Thus, both the angular frequency ù(k) and the phase velocity v ph are
dependent on the choice of the zero-level of the potential energy and are
therefore arbitrary; neither has a physical meaning for a wave packet represent-
ing a particle.
Since the parameter ã is non-vanishing, the wave packet will disperse with
time as indicated by equation (1.28). For a gaussian pro®le, the absolute value
of the wave packet is given by equation (1.31) with ã given by (1.43). We note
ÿ1
that ã is proportional to m , so that as m becomes larger, ã becomes smaller.
Thus, for heavy particles the wave packet spreads slowly with time.
As an example, the value of ã for an electron, which has a mass of
2 ÿ1
9:11 3 10 ÿ31 kg, is 5:78 3 10 ÿ5 m s . For a macroscopic particle whose
mass is approximately a microgram, say 9:11 3 10 ÿ10 kg in order to make the
2 ÿ1
calculation easier, the value of ã is 5:78 3 10 ÿ26 m s . The ratio of the
21
macroscopic particle to the electron is 10 . The time dependence in the
dispersion terms in equations (1.31) occurs as the product ãt. Thus, for the
same extent of spreading, the macroscopic particle requires a factor of 10 21
longer than the electron.

1.5 Heisenberg uncertainty principle

Since a free particle is represented by the wave packet Ø(x, t), we may regard
the uncertainty Äx in the position of the wave packet as the uncertainty in the
position of the particle. Likewise, the uncertainty Äk in the wave number is
related to the uncertainty Äp in the momentum of the particle by Äk ˆ Äp=".
The uncertainty relation (1.23) for the particle is, then
ÄxÄp >" (1:44)
This relationship is known as the Heisenberg uncertainty principle.
The consequence of this principle is that at any instant of time the position