Page 31 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

22 The wave function

of the particle is de®ned only as a range Äx and the momentum of the particle
is de®ned only as a range Äp. The product of these two ranges or `uncertain-
ties' is of order " or larger. The exact value of the lower bound is dependent on
how the uncertainties are de®ned. A precise de®nition of the uncertainties in
position and momentum is given in Sections 2.3 and 3.10.
The Heisenberg uncertainty principle is a consequence of the stipulation that
a quantum particle is a wave packet. The mathematical construction of a wave
packet from plane waves of varying wave numbers dictates the relation (1.44).
It is not the situation that while the position and the momentum of the particle
are well-de®ned, they cannot be measured simultaneously to any desired degree
of accuracy. The position and momentum are, in fact, not simultaneously
precisely de®ned. The more precisely one is de®ned, the less precisely is the
other, in accordance with equation (1.44). This situation is in contrast to
classical-mechanical behavior, where both the position and the momentum can,
in principle, be speci®ed simultaneously as precisely as one wishes.
In quantum mechanics, if the momentum of a particle is precisely speci®ed
so that p ˆ p 0 and Äp ˆ 0, then the function A( p)is
A(p) ˆ ä( p ÿ p 0 )

The wave packet (1.37) then becomes

1 1 i( pxÿEt)=" 1 i( p 0 xÿEt)="
Ø(x, t) ˆ p   ä(p ÿ p 0 )e dp ˆ p  e

2ð"
2ð" ÿ1
which is a plane wave with wave number p 0 =" and angular frequency E=".
Such a plane wave has an in®nite value for the uncertainty Äx. Likewise, if the
position of a particle is precisely speci®ed, the uncertainty in its momentum is
in®nite.
Another Heisenberg uncertainty relation exists for the energy E of a particle
and the time t at which the particle has that value for the energy. The
uncertainty Äù in the angular frequency of the wave packet is related to the
uncertainty ÄE in the energy of the particle by Äù ˆ ÄE=", so that the
relation (1.25) when applied to a free particle becomes
ÄEÄt >" (1:45)
Again, this relation arises from the representation of a particle by a wave
packet and is a property of Fourier transforms.
The relation (1.45) may also be obtained from (1.44) as follows. The
uncertainty ÄE is the spread of the kinetic energies in a wave packet. If Äp is
small, then ÄE is related to Äp by

p 2 p
ÄE ˆ Ä ˆ Äp (1:46)
2m m