Lecture 5: Coordinate and Momentum
Representations
•
We will start by considering the quantum
description of the motion of a particle in
one dimension.
•
In classical mechanics, the state of the
particle is given by its position and
momentum coordinates,
x
and
p
.
•
In quantum mechanics, we will consider
position and momentum as observables and
therefore represent them by Hermitian
operators,
X
and
P
, respectively.
•
According to the Seventh Postulate, these
two operators obey the commutation
relation:
[
]
h
i
P
X
=
,
Incompatible Observables
•
If two operators do not commute, then an
eigenstate of one cannot be an eigenstate of the
other.
Proof:
–
Let
[
A,B
]=
M
–
Assume that

a,b
!
is simultaneously an eigenstate of
A
and
B
:
–
Operate on

a,b
!
with
M
:
–
If
A
and
B
commute (
M
=0
) then

a,b
!"
0
A
and
B
are then called
‘compatible’
–
If
A
and
B
do not commute (M!= 0) then there is no
solution (i.e.

a,b
!
=0
is only solution)
A
and
B
are then called
‘incompatible’
b
a
b
b
a
B
b
a
a
b
a
A
,
,
,
,
=
=
(
)
(
)
0
,
)
(
,
,
,
,
,
,
,
=
!
=
!
=
!
=
!
=
b
a
ab
ba
b
a
ab
b
a
ba
b
a
a
B
b
a
b
A
b
a
BA
b
a
AB
b
a
M