Operators
•
In QM, an operator is an object that acts on a ket,
transforming it into another ket
– Let
A
represent a generic operator
–
An operator is a linear map
A
:
H
!
H
A
|
"#
=
|
"
’
#
–
Operators are linear:
A
(
a
|
"
1
#
+b
|
"
2
#
)
=
aA
|
"
1
#
+
bA
|
"
2
#
•
a
and
b
are arbitrary c-numbers
Notation:
–
Generally, we will follow Cohen-Tannoudji, and
use capital letters for operators and lower-case
letters for c-numbers.
–
Another common convention is to distinguish
operators by giving them a ‘^’
•
I may use this occasionally
a
ˆ
!
ˆ
A
ˆ
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Matrix representations
–
Just as kets are vectors, operators are matrices
–
Let the set
{
|
1
#
,
|
2
#
,
|
3
#
,…,
|
M
#
}
be a set of orthogonal
unit vectors which spans an entire
M
-dimensional
Hilbert space
–
The c-number
$
j
|
"#
is thus the
j
th
component of the
vector
|
"#
–
Matrix Representation of an operator:
•
Start from the equation:
•
Hit it from the left with the bra
$
j
|
:
•
Insert `the identity’ between
A
and
|
"#
:
•
Use the replacements:
c
j
%
$
j
|
"
#
,
d
j
%
$
j
|
"
’
#
and
A
jk
%
$
j
|
A
|
k
#
to get:
–
This is just the standard formula for matrix multiplication:
!
!
A
=
'
!
!
A
j
j
=
"
!
!
k
k
A
j
j
M
k
"
=
=
#
1
k
M
k
jk
j
c
A
d
!
=
=
1
v
d
=
A
r
c
Defining states and operators
•
A state (vector) is specified by giving its
components in some physically meaningful
basis
•
An operator is defined by giving its matrix
elements in some physically meaningful basis
•
Operators and/or states can alternatively be
defined as the solution to a particular equation
–

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- Fall '08
- Moore,M
- Linear Algebra, mechanics, Hilbert space, Hermitian, Hermitian Operators
-
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