Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
It is also easy to show that
…A
~
B
~
† ‡ ˆ B
~ ‡ A
~
‡ :
…5:23†
For any jui; jvi, hvjB
~ ‡ and B
~
jvi is a pair of dual vectors; hujA
~ ‡ and A
~
jui is also a
pair of dual vectors. Thus we have
hjB v
‡ A ‡ jui ˆ fhjB v
‡ gfA ‡ juig ˆ ‰fhujA gfB jvigŠ*
~ ~ ~ ~ ~ ~
and therefore
ˆ hujA B ji* v ˆ hj…A v B † ‡ ji u
~ ~ ~ ~
…A
~
B
~
† ‡ ˆ B
~ ‡ A
~
‡ :
Hermitian operators
An operator H
~
that is equal to its adjoint, that is, that obeys the relation
H
~
ˆ H
~
‡
…5:24†
is called Hermitian or self-adjoint. And H
~
is anti-Hermitian if
H
~
ˆH
~
‡ :
Hermitian operators have the following important properties:
(1) The eigenvalues are real: Let H
~
be the Hermitian operator and let jvi be an
eigenvector belonging to the eigenvalue :
H
~
jvi ˆjvi:
By de®nition, we have
that is,
Since hvjvi 6ˆ0, we have
hjA v jiˆ v hjA v ji*; v
~ ~
…* † hvv
jiˆ 0:
* ˆ :
(2) Eigenvectors belonging to di€erent eigenvalues are orthogonal: Let jui and
jvi be eigenvectors of H belonging to the eigenvalues and respectively:
~
H jui ˆjui; H jvi ˆ jvi:
~ ~
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