Lect3_Operators - Notation Operators In QM an operator is...

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

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
–