Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL OPERATORS
Then
That is,
But 6ˆ , so that
u h jH
~
… †hvjui ˆ0
jiˆ v hjH v
~
hvjui ˆ0:
jui*:
…since * ˆ †:
(3) The set of all eigenvectors of a Hermitian operator forms a complete set:
The eigenvectors are orthogonal, and since we can normalize them, this
means that the eigenvectors form an orthonormal set and serve as a basis
for the vector space.
Unitary operators
A linear operator U is unitary if it preserves the Hermitian character of an
~
operator under a similarity transformation:
…U A U 1 † ‡ 1 ˆ U A U ;
~ ~ ~ ~ ~ ~
where
A ‡ ˆ A :
~ ~
But, according to Eq. (5.23)
…U A U 1 † ‡ ˆ…U 1 † ‡ A U ‡ ;
~ ~ ~ ~ ~ ~
thus, we have
…U 1 † ‡ A U ‡ ˆ U A U 1 :
~ ~ ~ ~ ~ ~
Multiplying from the left by U ‡ and from the right by U , we obtain
~ ~
U ‡ …U 1 † ‡ A U ‡ U ˆ U ‡U A ;
~ ~ ~ ~ ~ ~ ~ ~
this reduces to
A
~
…U
~ ‡ U
~
†ˆ…U
~
‡U
~
†A
~
;
since
U
~ ‡ …U
~ 1 † ‡ ˆ…U
~ 1 U
~
† ‡ ˆ E
~
:
Thus
U
~ ‡ U
~
ˆ E
~
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