Page 75 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

66 General principles of quantum theory
x y
x
e x‡ y ˆ e e 6ˆ e ‡ e y
e cx 6ˆ ce x
2
2
2
2
[x ‡ y] ˆ x ‡ 2xy ‡ y 6ˆ x ‡ y 2
2
[c(x ‡ y)] 6ˆ c[x ‡ y] 2
^
^
^
The operator C is the sum of the operators A and B if
^
^
^
^
^
Cø ˆ (A ‡ B)ø ˆ Aø ‡ Bø
^
^
^
The operator C is the product of the operators A and B if
^
^ ^
^^
Cø ˆ ABø ˆ A(Bø)
^
^
where ®rst B operates on ø and then A operates on the resulting function.
Operators obey the associative law of multiplication, namely
^ ^ ^
^^ ^
A(BC) ˆ (AB)C
^
^ 2
Operators may be combined. Thus, the square A of an operator A is just the
^^
product AA
^^
^ 2
^ ^
A ø ˆ AAø ˆ A(Aø)
^
Similar de®nitions apply to higher powers of A. As another example, the
differential equation
2
d y
2
‡ k y ˆ 0
dx 2
^ 2
^ 2
2
2
may be written as (D ‡ k )y ˆ 0, where the operator (D ‡ k ) is the sum of
x x
2
^ 2
the two product operators D and k .
x
^^
^
^
In multiplication, the order of A and B is important because ABø is not
^
^
^ ^
^
necessarily equal to BAø. For example, if A ˆ x and B ˆ D x , then we have
^^ ^ ^ ^ ^
ABø ˆ xD x ø ˆ x(dø=dx) while, on the other hand, BAø ˆ D x (xø) ˆ
^
^
^ ^
ø ‡ x(dø=dx). The commutator of A and B, written as [A, B], is an operator
de®ned as
^ ^
^^
^ ^
[A, B] ˆ AB ÿ BA (3:3)
^ ^
^^
^ ^
^ ^
from which it follows that [A, B] ˆÿ[B, A]. If ABø ˆ BAø, then we have
^
^
^ ^
^ ^
^^
AB ˆ BA and [A, B] ˆ 0; in this case we say that A and B commute.By
expansion of each side of the following expressions, we can readily prove the
relationships
^ ^ ^
^ ^ ^
^ ^ ^
[A, BC] ˆ [A, B]C ‡ B[A, C] (3:4a)
^ ^ ^
^ ^ ^
^^ ^
[AB, C] ˆ [A, C]B ‡ A[B, C] (3:4b)
^ ^
^
^
^^
The operator A is the reciprocal of B if AB ˆ BA ˆ 1, where 1 may be
^
^ ÿ1
regarded as the unit operator, i.e., `multiply by unity'. We may write A ˆ B