Chem. 540
Nancy Makri
Dirac braket notation
The symbol
n
(or
n
Ψ
) is called a “ket” and
denotes the state described by the wavefunction
n
Ψ
.
The complex conjugate of the wavefunction,
n
Ψ
, is denoted by the “bra”
n
(or
n
Ψ
).
The ket
denotes a state in the most abstract form, without reference to a particular representation.
When we put a bra together with a ket, with an operator in the middle, we imply integration over
all space:
ˆ
ˆ
ˆ
( )
( )
or
n
m
n
m
d
A
A
n A m
Ψ
Ψ
≡
Ψ
Ψ
∫
r
r
r
Leaving out the operator implies the identity operator, i.e.,
( )
( )
m
n
m
n
n
m
d
Ψ
Ψ
≡
Ψ
Ψ
= Ψ
Ψ
∫
r
r
r
m
n
Ψ
Ψ
is the amplitude for a particle in state
n
Ψ
to be also in state
m
Ψ
.
This is also known
as the overlap of these two states.
In this notation, the condition for an operator to be hermitian is
ˆ
ˆ
m
n
n
m
A
A
Ψ
Ψ
=
Ψ
Ψ
.
Remarks
n
m
Φ
Φ
is a scalar (i.e., a number).
m
n
Φ
Φ
is an operator, because it can operate on
Ψ
to give
(
29
,
m
n
m
n
α
α
Φ
Φ
Ψ =
Φ
= Φ
Ψ
ˆ
n
n
n
P
≡ Φ
Φ
is a projection operator.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Mccall
 Particle

Click to edit the document details