Mathematical Methods for Physicists: A concise introduction - Site Map

THE ALGEBRA OF LINEAR OPERATORS
Similarly
2 ˆh2jL
~
j1i 1 ‡h2jL
~
j2i 2 ˆ 21 1 ‡ 22 2 ‡;
3 ˆh3jL
~
j1i 1 ‡h3jL
~
j2i 2 ˆ 31 1 ‡ 32 2 ‡:
In general, we have
where
i ˆ X
ij j ;
j
ij ˆ hjL i ji: j
~
…5:17†
Consequently, in terms of the vectors j1i; j2i; ... as a basis, operator L is represented
by the matrix whose elements are ij .
~
A matrix representing L can be found by using any basis, not necessarily an
~
orthonormal one. Of course, a change in the basis changes the matrix representing
L .
~
The algebra of linear operators
Let A and B be two operators de®ned in a linear vector space V of vectors ji. The
~ ~
equation A ˆ B will be understood in the sense that
~ ~
A jiˆB ji for all ji2V:
~ ~
We de®ne the addition and multiplication of linear operators as
C ˆ A ‡ B and D ˆ A B
~ ~ ~ ~ ~ ~
if for any ji
C
~
jiˆ…A
~
‡ B
~
†j i ˆ A
~
ji‡ B
~
ji;
D jiˆ…A B †j i ˆ A …B ji†:
~ ~ ~ ~ ~
Note that A ‡ B and A B are themselves linear operators.
~ ~ ~ ~
Example 5.10
(a) …A B †…ji‡ u ji†ˆ v A ‰…B jui†‡ …B jvi†Š ˆ …A B † jui‡ …A B † ji; v
~ ~ ~ ~ ~ ~ ~ ~ ~
(b) C …A ‡ B † jiˆC v …A ji‡B v ji†ˆC v A ji‡C B ji,
~ ~ ~ ~ ~ ~ ~ ~ ~ ~
which shows that
C
~
…A
~
‡ B
~
†ˆC
~
A
~
‡ C
~
B
~
:
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