Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL OPERATORS
The inverse of a product of operators is the product of the inverse in the reverse
order
…A
~
B
~
† 1 ˆ B
~ 1 A
~
1 :
…5:20†
The proof is straightforward: we have
A B …A B † 1 ˆ E :
~ ~ ~ ~ ~
Multiplying successively from the left by A 1 and B 1 , we obtain
~ ~
…A B † 1 ˆ B 1 1
A ;
~ ~ ~ ~
which is identical to Eq. (5.20).
The adjoint operators
Assuming that V is an inner-product space, then the operator X
~
relation
hujX jiˆ v hjA v jui* for any jui; jvi 2V
~
~
satisfying the
is called the adjoint operator of A
~
and is denoted by A
~ ‡ . Thus
hujA ‡ ji v hjA v ji* u for any jui; jvi 2V: …5:21†
~ ~
We ®rst note that hjA
~ ‡ is a dual vector of A
~
ji. Next, it is obvious that
…A
~ ‡ † ‡ ˆ A
~
:
…5:22†:
To see this, let A
~ ‡ ˆ B
~
, then (A
~ ‡ † ‡ becomes B
~ ‡ , and from Eq. (5.21) we ®nd
But
Thus
from which we ®nd
hjB v
‡ jui ˆ hujB ji*; v for any jui; jvi 2V:
~ ~
hujB ji* v ˆ hujA ‡ ji* v ˆ hjA v jui:
~ ~ ~
hjB v
‡ jiˆ u hujB ji* v ˆ hjA v jui
~ ~ ~
…A
~ ‡ † ‡ ˆ A
~
:
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