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Hermitian Operators
•
Definition: an operator is said to be Hermitian if
it satisfies:
A
†
=A
–
Alternatively called ‘self adjoint’
–
In QM we will see that all observable properties
must be represented by Hermitian operators
•
Theorem: all eigenvalues of a Hermitian
operator are real
–
Proof:
•
Start from Eigenvalue Eq.:
•
Take the H.c. (of both sides):
•
Use
A
†
=A:
•
Combine to give:
•
Since
!
a
m

a
m
"
#
0 it follows that
m
m
m
a
a
A
a
!
=
†
m
m
m
a
a
A
a
!
=
m
m
m
m
m
m
m
m
a
a
a
a
a
a
a
A
a
=
=
!
m
m
a
a
=
!
m
m
m
a
a
a
A
=
Eigenvectors of a Hermitian operator
–
Note: all eigenvectors are defined only up to a
multiplicative cnumber constant
•
Thus we can choose the normalization
!
a
m

a
m
"
=1
•
THEOREM: all eigenvectors corresponding to
distinct eigenvalues are orthogonal
–
Proof:
•
Start from eigenvalue equation:
•
Take H.c. with m
$
n:
•
Combine to give:
•
This can be written as:
•
So either
a
m
= a
n
in which case they are not
distinct, or
!
a
m

a
n
"
=0
, which means the
eigenvectors are orthogonal
(
)
(
)
m
m
m
a
c
a
a
c
A
=
!
m
m
m
a
a
a
A
=
m
m
m
a
a
a
A
=
n
n
n
a
a
A
a
=
m
n
m
m
n
n
m
n
a
a
a
a
a
a
a
A
a
=
=
0
)
(
=
!
m
n
m
n
a
a
a
a
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View Full DocumentCompleteness of Eigenvectors of a
Hermitian operator
•
THEOREM: If an operator in an Mdimensional
Hilbert space has M distinct eigenvalues (i.e.
no degeneracy), then its eigenvectors form a
`complete set’ of unit vectors (i.e a complete
‘basis’)
–
Proof:
M orthonormal vectors must span an
Mdimensional space.
•
Thus we can use them to form a
representation of the identity operator:
Degeneracy
•
Definition: If there are at least two linearly
independent eigenvectors associated with the
same eigenvalue, then the eigenvalue is
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 Spring '09
 Lee
 mechanics

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