Page 111 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

102 General principles of quantum theory

equivalent procedure. The relation (3.82) is consistent with (1.44), which is
based on the Fourier transform properties of wave packets. The difference
between the right-hand sides of (1.44) and (3.82) is due to the precise de®nition
(3.75) of the uncertainties in equation (3.82).
Similar applications of equation (3.81) using the position±momentum pairs
y, ^ p y and z, ^ p z yield
" "
ÄyÄp y > , ÄzÄp z >
2 2
Since x commutes with the operators ^ p y and ^ p z , y commutes with ^ p x and ^ p z ,
and z commutes with ^ p x and ^ p y , the relation (3.81) gives
Äq i Äp j ˆ 0, i 6ˆ j

where q 1 ˆ x, q 2 ˆ y, q 3 ˆ z, p 1 ˆ p x , p 2 ˆ p y , p 3 ˆ p z . Thus, the position
coordinate q i and the momentum component p j for i 6ˆ j may be precisely
determined simultaneously.

Minimum uncertainty wave packet
The minimum value of the product ÄAÄB occurs for a particular state Ø for
which the relation (3.81) becomes an equality, i.e., when
1 ^ ^
ÄAÄB ˆ jh[A, B]ij (3:83)
2
According to equation (3.79), this equality applies when
^
^
[A ÿhAi‡ ië(B ÿhBi)]Ø ˆ 0 (3:84)
^
where ë is given by (3.80). For the position±momentum example where A ˆ x
^
and B ˆÿi" d=dx, equation (3.84) takes the form

d i
ÿi" ÿhp x i Ø ˆ (x ÿhxi)Ø
dx ë
for which the solution is
2
Ø ˆ ce ÿ(xÿhxi) =2ë" ih p x ix=" (3:85)
e
where c is a constant of integration and may be used to normalize Ø. The real
constant ë may be shown from equation (3.80) to be
" 2(Äx) 2
ë ˆ ˆ
2(Äp x ) 2 "
where the relation ÄxÄp x ˆ "=2 has been used, and is observed to be positive.
Thus, the state function Ø in equation (3.85) for a particle with minimum
position±momentum uncertainty is a wave packet in the form of a plane wave
exp[ihp x ix="] with wave number k 0 ˆhp x i=" multiplied by a gaussian
modulating function centered at hxi. Wave packets are discussed in Section