Higher Order Perturbation Theory
Michael Fowler 03/07/06
The Interaction Representation
Recall that in the first part of this course sequence, we discussed the Schrödinger and Heisenberg
representations of quantum mechanics
here
.
In the Schrödinger representation, the operators are
timeindependent (except for explicitly timedependent potentials) the kets representing the
quantum states develop in time.
In the Heisenberg representation, the kets stay the same, the
time dependence is in the operators.
These differing representations describe the same physics—
matrix elements of operators between kets must be the same in both.
The most natural to use
depends on the problem at hand. In the classical limit, for example, the Heisenberg operators
have the time dependence of the corresponding classical operators.
In fact, for perturbation theory problems with a timedependent potential, an intermediate
representation, the
interaction representation
, is very convenient.
Using a subscript
S
to denote
the Schrödinger representation,
( )
( )
( )
(
)
( )
0
,
S
S
S
S
S
S
d
i
t
H
t
H
V
t
dt
ψ
ψ
ψ
=
=
+
=
t
we define the
interaction representation
by the unitary transformation
( )
( )
0
/
S
iH t
I
S
t
e
t
ψ
ψ
=
=
so the interaction representation kets and the Schrödinger representation kets coincide at
t
= 0,
and if the interaction were zero, the interaction representation kets would be constant in time,
like those in the Heisenberg representation.
For nonzero
( )
V t
, then, the time development of the interaction representation kets is entirely
due to
( )
V t
, and is easily found by differentiating both sides of the equation:
( )
( )
( )
( )
( )
(
)
( )
( )
( )
0
0
0
0
0
/
0
/
/
0
0
/
/
( )
( )
,
S
S
S
S
S
iH t
I
I
S
iH t
iH t
I
S
S
iH t
iH t
S
I
I
I
d
d
i
t
H
t
e
i
t
dt
dt
H
t
e
H
V
t
e
t
e
V
t e
t
V
t
t
ψ
ψ
ψ
ψ
ψ
ψ
ψ
−
−
= −
+
= −
+
+
=
=
=
=
=
=
=
=
=
I
where we have introduced the interaction representation operator
V
I
(
t
), defined by
( )
( )
0
0
/
/
.
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 Fall '07
 MichaelFowler
 mechanics, Schrödinger Representation, interaction representation, interaction representation kets

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