Page 54 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.3 Expectation values of dynamical quantities 45

2
1
2
1 @ Ø @Ø @Ø @Ø 1 @Ø @ Ø
dx ˆ ÿ dx
@x 2 @x @x @x @x @x 2
ÿ1 ÿ1 ÿ1
1
3

2
@Ø @Ø @ Ø 1 @ Ø
ˆ ÿ Ø ‡ Ø dx
@x @x @x 2 ÿ1 ÿ1 @x 3
The integrated part vanishes because Ø and @Ø=@x vanish at ( ) in®nity. The
remaining integral cancels the second integral on the right-hand side of
equation (2.24), leaving the ®nal result

dhpi dV
ˆÿ ˆhFi (2:25)
dt dx
where equation (2.1) has been used. Equation (2.25) is the quantum analog of
Newton's second law of motion, F ˆ ma, and is in agreement with the
correspondence principle.

Heisenberg uncertainty principle
Using expectation values, we can derive the Heisenberg uncertainty principle
introduced in Section 1.5. If we de®ne the uncertainties Äx and Äp as the
standard deviations of x and p, as used in statistics, then we have

2 1=2
Äx ˆh(x ÿhxi) i
2 1=2
Äp ˆh(p ÿhpi) i
The expectation values of x and of p at a time t are given by equations (2.13)
and (2.18), respectively. For the sake of simplicity in this derivation, we select
the origins of the position and momentum coordinates at time t to be the
centers of the wave packet and its Fourier transform, so that hxiˆ 0 and
hpiˆ 0. The squares of the uncertainties Äx and Äp are then given by

1
2 2
(Äx) ˆ x Ø Ø dx
ÿ1
" # 1
2
2
2
2
" 1 @ Ø " @Ø " 1 @Ø @Ø
2
(Äp) ˆ Ø dx ˆ Ø ÿ dx
i @x 2 i @x i @x @x
ÿ1 ÿ1
ÿ1

1 ÿ" @Ø " @Ø
ˆ dx
i @x i @x
ÿ1
2
where the integrated term for (Äp) vanishes because Ø goes to zero as x
approaches ( ) in®nity.
2
The product (ÄxÄp) is