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 cnumbers
Notation:
–
Generally, we will follow CohenTannoudji, and
use capital letters for operators and lowercase
letters for cnumbers.
–
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 cnumber
$
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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