6.2 Quantization of the Electromagnetic Field
173
We can now use the expansions (
6.17
) and (
6.19
) as well as (
6.18
) and (
6.20
) to
define the electric and magnetic field operators
ˆ
E
and
ˆ
B
in terms of the operators
ˆ
A
k
,
A
†
k
and thus of
a
k
α
,
a
†
k
α
:
ˆ
E
=
k
ˆ
E
k
e
i
k
·
x
+
ˆ
E
†
k
e
−
i
k
·
x
;
ˆ
B
=
k
ˆ
B
k
e
i
k
·
x
+
ˆ
B
†
k
e
−
i
k
·
x
.
(6.46)
with:
ˆ
E
k
=
2
α
=
1
ˆ
E
k
α
u
k
,α
=
2
α
=
1
i
ω
k
2
V
a
k
α
u
k
,α
;
ˆ
B
k
=
2
α
=
1
ˆ
B
k
α
u
k
,α
=
n
k
×
ˆ
E
k
.
(6.47)
We are now able to compute the Hamiltonian operator
ˆ
H
following the same deriva-
tion as in the classical case. Care, however, has to be used in deriving the operator
versions of equations (
6.27
) and (
6.28
) from (
6.26
) since, as opposed to the corre-
sponding classical quantities which were just numbers, the operators
ˆ
E
k
and
ˆ
E
†
k
, as
well as their magnetic counterparts, no longer commute. As a consequence of this,
in writing the expression for the Hamiltonian, we should keep the order of factors
in each product and thus, instead of a sum over
ˆ
A
k
ˆ
A
†
k
, we would find a sum over
1
2
(
ˆ
A
k
ˆ
A
†
k
+
ˆ
A
†
k
ˆ
A
k
)
. The Hamiltonian operator then reads:
ˆ
H
=
k
α
ω
k
2
a
k
,α
a
†
k
α
+
a
†
k
α
a
k
α
=
k
α
ˆ
N
k
,α
+
1
2
ω
k
,
(6.48)
where
ˆ
N
k
,α
≡
a
†
k
,α
a
k
,α
,
(6.49)
The operator
ˆ
H
, in terms of the canonical operators, has the same form as in (
6.34
):
ˆ
H
=
1
2
k
α
(
ˆ
P
k
,α
)
2
+
ω
k
(
ˆ
Q
k
,α
)
2
.
(6.50)
Equations (
6.48
) and (
6.50
) describe the Hamiltonian operator associated with the
system of infinitely many quantum harmonic oscillators
(
k
, α)
defined in the previous
section. The quantities
a
k
,α
and
a
†
k
,α
are indeed nothing but the annihilation and
creation operators associated with the quantum oscillator
(
k
, α)
, which are useful in
constructing the corresponding quantum states. It is now straightforward to determine
the expression for the momentum operator, by using (
6.38
) and (
6.48
):
ˆ
P
=
1
c
V
d
3
xE
×
B
=
k
α
k
ˆ
N
k
,α
+
1
2
.
(6.51)
Both
H
and
ˆ
P
are expressed in terms of the
occupation number
operators
ˆ
N
k
,α
(
6.49