Page 109 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

100 General principles of quantum theory
X X X X X
^ 2
hBiˆ c c i hâ k jBjâ i iˆ c c i â i ä ki ˆ jc i j â i
k k
i k i k i
where (3.73b) has been used. Thus, a measurement of B yields one of the many
2
values â i with a probability jc i j . There is no way to predict which of the values
â i will be obtained and, therefore, the observables A and B cannot both be
determined concurrently.
For a system in an arbitrary state Ø, neither of the physical observables A
^
^
and B can be precisely determined simultaneously if A and B do not commute.
Let ÄA and ÄB represent the width of the spread of values for A and B,
2
respectively. We de®ne the variance (ÄA) by the relation
2 ^ 2
(ÄA) ˆh(A ÿhAi) i (3:75)
that is, as the expectation value of the square of the deviation of A from its
mean value. The positive square root ÄA is the standard deviation and is called
the uncertainty in A. Noting that hAi is a real number, we can obtain an
2
alternative expression for (ÄA) as follows:
2 ^ 2 ^ 2 ^ 2
(ÄA) ˆh(A ÿhAi) iˆhA ÿ 2hAiA ‡hAi i
^ 2 2 ^ 2 2
ˆhA iÿ 2hAihAi‡hAi ˆhA iÿhAi (3:76)
2
Expressions analogous to equations (3.75) and (3.76) apply for (ÄB) .
^
^
^
Since A and B do not commute, we de®ne the operator C by the relation
^ ^
^
^^
^ ^
[A, B] ˆ AB ÿ BA ˆ iC (3:77)
^
The operator C is hermitian as discussed in Section 3.3, so that its expectation
^
^
value hCi is real. The commutator of A ÿhAi and B ÿhBi may be expanded
as follows
^
^
^
^
^
^
[A ÿhAi, B ÿhBi] ˆ (A ÿhAi)(B ÿhBi) ÿ (B ÿhBi)(A ÿhAi)
^
^ ^
^^
ˆ AB ÿ BA ˆ iC (3:78)
where the cross terms cancel since hAi and hBi are numbers and commute with
^
^
the operators A and B. We use equation (3.78) later in this section.
We now introduce the operator
^
^
A ÿhAi‡ ië(B ÿhBi)
where ë is a real constant, and let this operator act on the state function Ø
^
^
[A ÿhAi‡ ië(B ÿhBi)]Ø
The scalar product of the resulting function with itself is, of course, always
positive, so that
^
^
^
^
h[A ÿhAi‡ ië(B ÿhBi)]Øj[A ÿhAi‡ ië(B ÿhBi)]Øi > 0 (3:79)
Expansion of this expression gives